Document [original]

Dynamic Hedging in Illiquid Financial Markets

vorgelegt von

Dipl.-Math.oec., M.Sc.

Moritz Voß

geboren in Düsseldorf

von der Fakultät II – Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

Dr. rer. nat.

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Fredi Tröltzsch (TU Berlin)

Gutachter: Prof. Dr. Peter Bank (TU Berlin)

Gutachter: Prof. Dr. Ulrich Horst (Humboldt-Universität zu Berlin)

Gutachter: Prof. H. Mete Soner, Ph.D. (ETH Zürich)

Tag der wissenschaftlichen Aussprache: 3. Juli 2017

Berlin 2017

Abstract

In this thesis, we address the problem of constructing effective hedging strategies

against the financial risk of writing a contingent claim in an illiquid financial market.

Mathematically, this amounts to study various stochastic optimal control problems

with suitable nonlinear dynamics. We introduce a price impact model which ac-

counts for finite market depth, market tightness and finite resilience whose coupled

bid- and ask-price dynamics induce convex liquidity costs. We provide existence of

an optimal solution to the classical problem of maximizing expected utility from

terminal liquidation wealth at some finite planning horizon. In a specific configura-

tion of our model, it turns out that the resulting singular optimal stochastic control

problem reduces to a deterministic singular control problem. Rather than study-

ing the associated Hamilton-Jacobi-Bellmann PDE, we exploit convex analytic and

calculus of variations techniques which allow us to construct the solution explic-

itly and to describe analytically the free boundaries of the action- and non-action

regions in the underlying state space.

In the second part, we relate the optimal singular stochastic control problem of

utility-based hedging in our original illiquid market model to a considerably simpler

classical linear quadratic stochastic optimal tracking problem of a frictionless hedg-

ing strategy with constant coefficients. We solve this problem explicitly for general

predictable target hedging strategies. The consideration of general predictable ref-

erence processes is made possible by the use of a convex analytic approach along

the lines of Pontryagin’s maximum principle instead of the more common dynamic

programming methods. From a financial point of view, our results allow for an

intuitively appealing interpretation and yield sensible hedging strategies in illiquid

markets.

In the third part, we provide a probabilistic formulation of and solution to a

more general class of linear quadratic stochastic tracking problems with stochas-

tic coefficients and stochastic terminal state constraint. Proposing a suitable time

consistent approximation of the optimization problem allows us to tackle the fi-

nal state constraint which induces singular terminal conditions on the underlying

backward stochastic differential equations (BSDEs). Our approach also allows us

to provide necessary and sufficient conditions under which the constrained stochas-

tic optimization problem admits a finite value. We show that the optimal policy is

given by a similar form to the one obtained in the constant coefficient case.

iii

Zusammenfassung

Diese Arbeit beschäftigt sich mit der Konstruktion effektiver Absicherungsstrategi-

en gegen die finanziellen Risiken, die beim Verkauf von Finanzoptionen in illiquiden

Finanzmärkten entstehen. Mathematisch bedeutet dies die Analyse verschiedener

stochastischer optimaler Kontrollprobleme mit nicht-linearen Dynamiken. Wir füh-

ren ein Preiseinflussmodell ein, das sogenannte endliche Markttiefe, -enge sowie

-elastizität berücksichtigt und dessen gekoppelten Dynamiken der Geld- und Brief-

kurse konvexe Liquiditätskosten erzeugen. Wir zeigen die Existenz einer optimalen

Lösung für das klassische Nutzenmaximierungsproblem des erwarteten Endvermö-

gens bei endlichem Investitionszeitraum. In einer bestimmten Modellkonfiguration

zeigt sich, dass sich das ergebende optimale singuläre stochastische Kontrollproblem

auf ein singuläres deterministisches Kontrollproblem zurückführen lässt. Anstelle

die dazugehörige Hamilton-Jacobi-Bellmann PDGL zu untersuchen, verfolgen wir

einen konvex-analytischen Variationsrechnungsansatz der uns schließlich erlaubt,

die Lösung explizit zu bestimmen und die freien Ränder der aktiven und passiven

Kontrollregionen im zugrundeliegenden Zustandsraum analytisch zu beschreiben.

Im zweiten Teil führen wir das optimale singuläre stochastische Kontrollpro-

blem zur Berechnung nutzenbasierter Absicherungsstrategien in unserem ursprüng-

lichen illiquiden Finanzmarktmodell auf ein erheblich einfacheres klassisches linear-

quadratisches stochastisches optimales Zielverfolgungsproblem einer friktionslosen

optimalen Absicherungsstrategie mit konstanten Koeffizienten zurück. Wir lösen

dieses Problem explizit für allgemeine vorhersehbare Absicherungsstrategien als

Zielstrategie. Die Betrachtung allgemeiner Referenzstrategien wird anstatt der üb-

licheren dynamischen Programmierungsmethoden durch einen konvex-analytischen

Ansatz ähnlich zu Pontryagins Maximumsprinzip ermöglicht. Aus finanztechnischer

Sicht erlauben unsere Resultate eine intuitiv ansprechende Interpretation und be-

schreiben sinnvolle Absicherungsmöglichkeiten in illiquiden Finanzmärkten.

Im dritten Teil stellen wir eine probabilistische Formulierung sowie Lösung einer

allgemeineren Klasse stochastischer linear-quadratischer optimaler Zielverfolgungs-

probleme mit stochastischen Koeffizienten sowie stochastischer Endbedingung vor.

Um die stochastische Endbedingung, die zu singulären Endwerten in den zugrunde-

liegenden stochastischen Rückwärtsgleichungen führt, mathematisch handhabbarer

zu machen, schlagen wir eine geeignete zeitkonsistente Approximation des Optimie-

rungsproblems vor. Unsere Vorgehensweise erlaubt die Bestimmung von notwendi-

gen und hinreichenden Bedingungen unter denen das Optimierungsproblem einen

endlichen Lösungswert besitzt. Zudem erweist sich, dass die optimale Lösung von

ähnlicher Form ist wie im vorherigen Fall mit konstanten Koeffizienten.

Acknowledgment

First and foremost, I would like to express my deepest and sincere gratitude to my

supervisor Prof. Dr. Peter Bank. Without his ideas and continuing support during

the last five years this thesis would clearly not have been possible. He was always

available for intense and inspiring discussions. His very broad and deep knowledge

about Stochastic Analysis and Financial Mathematics significantly promoted my

thesis in an invaluable way. In addition, he always gave me the possibility to attend

conferences and workshops which allowed me to present my research and to meet

recognized experts. Beyond my Ph.D. project, his personal mentoring and advise

as a Doktorvater helped me substantially to develop a well-rounded and confident

personality as a mathematician and researcher.

Secondly, I would like to express my gratitude to Prof. Dr. Mete Soner for the

very rewarding collaboration. It has been a great privilege to work with him. I

would also like to thank him for readily accepting to co-examine this thesis.

Moreover, I would like to thank Prof. Dr. Ulrich Horst for having given me

the opportunity to present and share my research with leading experts several

times during the Berlin-Princeton-Singapore Workshops. I am very grateful that

he readily accepted to be my co-examiner.

Furthermore, I would like to thank my friends and colleagues at and outside TU

Berlin. Special thanks go to David Beßlich, Thibaut Lux, Alexandros Saplaouras

and Benjamin Stemper for the very enjoyable atmosphere in our working group.

In addition, I would like to thank Paulwin Graewe and Oliver Janke for the great

time we had organizing the Doktorandentreffen in Berlin.

Also, I would like to gratefully acknowledge the Research Training Group 1845

for the excellent qualification program during my Ph.D. studies in Berlin.

Finally, I would like to thank my wonderful brother Timo and my mother for

their continuous encouragement during the last years.

vii

Contents

1 Introduction 1

2 Optimal Investment with Transient Price Impact 11

2.1 A price impact model . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Optimal investment problem . . . . . . . . . . . . . . . . . . . . . . 14

2.3 Case study: Illiquid Bachelier model with exponential utility . . . . 16

2.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 High Resilience Heuristics for Utility-Based Hedging 85

4 Hedging with Temporary Price Impact 91

4.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.3 Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 General Stochastic Linear Quadratic Control for Hedging with

Temporary Price Impact 115

5.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.2 Connection between stochastic LQ problems and BS(R)DEs . . . . . 118

5.3 Stochastic LQ problem with stochastic terminal state constraint . . 123

5.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

1 Introduction

The construction of hedging strategies against financial risks is one of the key

problems in Mathematical Finance. The seminal work of Black and Scholes (1973)

and Merton (1973) showed how this task can be carried out in an idealized friction-

less market by dynamically trading perfectly liquid assets. In recent years there has

been a growing awareness that these idealizations can lead to misguided hedging

strategies with no longer negligible costs; particularly when these prescribe a fast

reallocation of assets in short periods of time in the presence of liquidity frictions

like price impact. In practice, market liquidity of a traded risky asset is finite en-

tailing that agents typically trade at bid- and ask-prices with a non-zero spread

which are possibly also adversely affected by the turned over volume or speed of

current and past executed trades. This has spurred the development of new fi-

nancial models which take into account the impact of transactions on execution

prices.

In this thesis, we address the problem of constructing effective hedging strate-

gies against the financial risk of writing a contingent claim in an illiquid financial

market. In the first part, we introduce a price impact model which accounts for

finite market depth,market tightness and finite resilience; the three dimensions of

liquidity identified in the seminal work by Kyle (1985). Finite market depth de-

scribes the required size of the traded volume to change prices by one monetary

unit. It induces transient price impact on the execution prices which does not

vanish instantaneously but persists and decays over time at some finite resilience

rate. Market tightness represents the cost per share of turning around a position

and is introduced by the bid-ask spread. The coupled bid- and ask-price dynamics

of our price impact model induce convex liquidity costs on the trading strategies

which are allowed to be singular comprising non-infinitesimal block trades. One

economically appealing choice to devise hedging strategies for an option seller is

to use utility indifference principles. We provide existence of an optimal solution

to the corresponding problem of maximizing expected utility from terminal liqui-

dation wealth at some finite planning horizon. In its simplest version, our price

impact model is an illiquid variant of a Bachelier model with convex liquidity costs

which are levied on the agent’s trading strategies. We show that the resulting

singular optimal control problem reduces in this framework to a deterministic op-

timal tracking problem of the constant buy-and-hold Merton portfolio. Instead

of the more common dynamic programming methods which lead to the challenge

of solving a free boundary problem induced by a Hamilton-Jacobi-Bellmann par-

tial differential equation, we exploit a convex analytic approach. Deriving first

1 Introduction

order conditions in terms of the (infinite dimensional) subgradients of the convex

cost functional allows us to construct explicitly the solution to the singular con-

trol problem by calculus of variations. As a consequence, we are able to describe

analytically the free boundaries of the action- and non-action regions in the under-

lying three-dimensional state space. It turns out that, depending on the agent’s

investment horizon, her initial position in the risky asset as well as the size of the

initial market spread, optimal strategies exhibit a surprisingly rich phenomenology

of possible trajectories for the optimal share holdings. Introducing a short posi-

tion of the investor in some random contingent claim in such an illiquid Bachelier

market, optimal strategies to the corresponding utility maximization problem will

in general not be deterministic anymore. Moreover, the complexity of the deter-

ministic closed-form solutions without random endowment suggests that explicit

characterizations of optimal utility-indifference hedging strategies will be hard to

derive already in this most elementary setup.

In the second part of this thesis, we therefore suggest to reduce the optimal

singular stochastic control problem of utility-based hedging in our original illiquid

market model with transient price impact to a considerably simpler classical linear

quadratic optimal hedging problem with non-singular strategies accounting only for

temporary price impact. We explain heuristically how the latter can be asymptoti-

cally retrieved from the former in a high resilience regime. The resulting stochastic

linear quadratic control problem comes in form of an optimal tracking problem of

a frictionless hedging strategy in the presence of quadratic illiquidity costs which

are levied on the agent’s respective trading rates. This problem formulation allows

the option seller to choose a preferred, yet due to illiquidity frictions not imple-

mentable hedging strategy adopted from a frictionless setting as her traget strategy

which she seeks to track in order to manage her financial risk of writing the claim.

In a Bachelier type framework with constant volatility and constant temporary

price impact, we solve this problem explicitly for general predictable target hedg-

ing strategies. Again, instead of the more common dynamic programming methods,

our approach is convex analytic along the lines of Pontryagin’s maximum princi-

ple which allows us to consider fairly general and not only continuous diffusion-

or Markovian-type target processes. This is particularly important for hedging in

illiquid financial markets when the frictionless reference hedge portfolio prescribes

sizable instantaneous reallocations. Our main contribution is the interpretation of

the optimal tracking strategy. It turns out that the optimal policy does not trade

from its current position towards the current target position but towards an optimal

signal process representing a weighted average of expected future target positions.

An interesting consequence from a financial point of view is that this averaging

allows to understand how singularities in the frictionless reference strategy have

to be addressed in models with illiquidity frictions like temporary price impact: A

singularity in the frictionless hedge is anticipated and systematically smoothed out

when averaging the weighted future target positions. This yields particularly sen-

sible hedging strategies for illiquid markets. In addition, the optimal signal process

also makes transparent how the regularity and predictability of the traget strategy

determine the problem’s optimal value.

In the third part of this thesis, we incorporate stochastic temporary price impact

and stochastic volatility in the linear quadratic benchmark hedging problem. We

show that in this more general setting a similar key role is played by a generalized

version of this optimal signal process. It also allows us to address the optimal

tracking problem with an additional delicate random terminal state constraint. In

case of a possible, not necessarily almost sure occurence of specific market condi-

tions, the agent may require to drive her hedging portfolio imperatively towards

a certain exogenously prescribed random value at terminal time. The stochastic

final state constraint in the linear quadratic optimization problem leads to singu-

lar terminal conditions in the underlying coupled system of backward stochastic

(Ricatti) differential equations (BSRDEs) which characterize the optimal solution

in the unconstrained case. We propose a way how the technical difficulties due

to the random terminal state constraint can be resolved by formulating a suitable

unconstrained variant of the associated constrained control problem. Our approach

allows us to identify necessary and sufficient conditions under which the tracking

problem with stochastic terminal state constraint actually admits a finite value.

We provide a probabilistic solution to this problem which can be considered as the

natural generalized extension of the results from the preceding constant volatility

and constant temporary price impact framework.

Optimal investment with transient price impact: Singular control

via convex analysis (Chapter 2)

The classical Merton (1971,1969) problem of maximizing expected utility from

terminal wealth by dynamically trading in a financial market has by now been

intensively studied and well understood in models with market frictions like trans-

action costs. We refer to the recent survey by Muhle-Karbe et al. (2016) for an

overview. In contrast, less is known about utility maximization problems in illiquid

market models where the friction is induced by price impact.

The two most widely used price impact models go back to Almgren and Chriss

(2001) as well as Obizhaeva and Wang (2013), respectively: The model of Alm-

gren and Chriss (2001) is characterized by directly specifying functions describing

the temporary and permanent impacts of a given order on the price. The model

of Obizhaeva and Wang (2013) assumes that trading takes place in an idealized

block-shaped limit order book with transient price impact which does not vanish

instantaneously but persists and decays over time at some finite resilience rate; cf.,

e.g., the surveys by Gökay et al. (2011) and Gatheral and Schied (2013).

Within these two models, the vast majority of the existing literature is primarily

concerned with the problem of optimally executing an exogenously given order by

1 Introduction

some fixed or finite time horizon; cf., e.g., Almgren and Chriss (2001), Almgren

(2003), Schied and Schöneborn (2009), Obizhaeva and Wang (2013), Alfonsi et al.

(2010), Predoiu et al. (2011), Ankirchner et al. (2014), Graewe et al. (2015), Kruse

and Popier (2016a), Graewe and Horst (2016), Graewe et al. (2017), Becherer

et al. (2017a,b). However, regarding more complex optimization problems such

as optimal portfolio choice, explicit characterizations of optimal strategies seem to

have been quite elusive so far. This is notably the case for optimal investment

problems on a finite time horizon in a model setup which accounts for a bid-

ask spread and price impact that, rather than being purely temporary or fully

permanent as proposed by Almgren and Chriss (2001), is transient as in Obizhaeva

and Wang (2013).

Most of the currently available work on optimal portfolio choice problems in illiq-

uid markets focuses on models with purely temporary price impact à la Almgren

and Chriss (2001), i.e., with infinite resilience, zero bid-ask spread, and restricts

to long-term investors as, e.g., in Guasoni and Weber (2015,2016,2017) with

constant relative risk aversion, in Forde et al. (2016) with constant absolute risk

aversion or in Gârleanu and Pedersen (2013a,b) with mean-variance preferences.

In the latter papers, the authors also take into account finite resilience as proposed

by Obizhaeva and Wang (2013). For investors having a finite planning horizon but

still facing solely purely temporary price impact, asymptotic results have been ob-

tained by Moreau et al. (2017) in a general Markovian setup for optimal expected

utility from terminal wealth. These results are also used as a building block to de-

scribe asymptotically optimal trading strategies under highly resilient price impact

in Kallsen and Muhle-Karbe (2014) or in Ekren and Muhle-Karbe (2017) in the

setting of Gârleanu and Pedersen (2013a). In all the above cited papers, except

in Gârleanu and Pedersen (2013b, Section 1.3), trading strategies are confined to

be absolutely continuous.

The paper most closely related to our work presented in Chapter 2is Soner and

Vukelja (2016); cf. also the PhD thesis by Vukelja (2014). They adopt the model

from Roch and Soner (2013) without bid-ask spread in a Black-Scholes framework

with constant resilience and stochastic market depth proportional to the risky asset

price. Using a dynamic programming approach and the notion of viscosity solu-

tions, the authors analyze the problem of maximizing expected utility from terminal

liquidation wealth for CRRA investors with finite planning horizon. Compared to

our results, their more general framework comes at the cost that a characterization

of the optimal strategy is only possible numerically via a discrete time approxima-

tion scheme.

In Section 2.1, we propose a price impact model which is a variant of the idealized

block-shaped limit order book model by Obizhaeva and Wang (2013) which allows

for both selling and buying stock. Specifically, our model determines bid- and

ask-prices via a coupled system of controlled diffusions, giving us the possibility

to specify market depth, market tightness and resilience; cf. the seminal paper

by Kyle (1985). The coupled bid- and ask-price dynamics induce convex liquidity

costs on the trading strategies which are allowed to be singular comprising non-

infinitesimal block trades as in Obizhaeva and Wang (2013). Our model is closely

related to the one proposed in Roch and Soner (2013) which is an extension of the

illiquid market model approach introduced by Çetin et al. (2004) in the sense that

it additionally takes into account finite resilience and a bid-ask spread. In contrast,

recovery of the bid- and ask-prices in the absence of the investor’s trading activity

is captured in our model by a reversion of these prices to each other and not to

some auxiliary reference price. Moreover, our illiquidity parameters, i.e., market

depth and resilience, are simply constants preserving tractability of our model.

We study the problem of maximizing expected utility from terminal liquidation

wealth at some finite time horizon in Section 2.2. Existence of an optimal strategy

is provided in a general setup. In its simplest version, our price impact model is an

illiquid variant of a Bachelier model. In Section 2.3, we present a comprehensive

case study in this framework for an investor who exhibits constant absolute risk

aversion. It turns out that the resulting singular optimal control problem reduces to

a deterministic optimal tracking problem of the frictionless constant buy-and-hold

Merton portfolio where convex liquidity costs are levied on the investor’s trading

activity. Using methods from convex analysis and calculus of variations, we provide

an explicit description of the three-dimensional state space which separates into a

buying-, selling and a no-trading region for the optimally controlled dynamics of the

spread and the risky asset holdings with respect to the remaining time to maturity;

cf. the numerical illustration in Section 2.3.5. The derived convex-analytic first

order conditions take the form of so-called “flat-off” conditions for the optimal

singular controls in terms of the infinite dimensional subgradients of the convex

cost functional. In fact, they allow for an explicit construction of optimal policies

as well as an analytic description of the free boundaries of the action and non-

action regions in the state space. A similar stochastic singular control problem is

studied in Horst and Naujokat (2014). The authors derive a suitable version of

the stochastic maximum principle and characterize optimal strategies in terms of

forward-backward stochastic differential equations. These equations induce likewise

buy-, sell- and no-trade regions in the underlying state space. The additional

randomness makes it difficult though to describe these regions analytically.

Our explicit results from Section 2.3 allow us to gain further insight into the

structure of optimal investment strategies in illiquid markets and make transpar-

ent how the optimizer has to comprise several aspects. As expected by the results

in Guasoni and Weber (2015,2016,2017), Forde et al. (2016), Gârleanu and Peder-

sen (2013a,b) and Moreau et al. (2017) it is optimal to trade towards the frictionless

constant Merton portfolio while taking into account the initial bid-ask spread at the

same time. Since liquidation is costly, the optimizer also has to unwind in an opti-

mal manner his accrued position in the risky asset when approaching maturity. In

fact, the emergence and interplay of these issues renders our problem substantially

1 Introduction

different from the infinite horizon and zero spread frameworks considered in the

papers cited above. The interaction of market tightness, finite resilience, desired

position targeting and optimal liquidation at a finite time horizon leads already

in this elementary illiquid Bachelier setup to a surprisingly rich phenomenology of

possible trajectories for the optimal share holdings; cf. Figure 1.1 below.

012345 τ

-5

Figure 1.1: Evolution of optimal share holdings of the risky asset for different time

to maturities τ, initial positions φin the asset (represented by the dots)

and initial values of the market spread. The final share holdings are

always zero. The grey line represents the constant Merton position. For

example, the blue policy decomposes into a waiting-, buying-, waiting-

and selling part; the black policy is of buy-and-hold type with initial and

final block trades; the pink policy with initial five shares does not trade

at all and unwinds at the end; the red policy is an optimal liquidation

strategy similar to those computed in Obizhaeva and Wang (2013) with

zero initial spread.

From transient to temporary price impact: Heuristic high

resilience asymptotics (Chapter 3)

Concerning our goal of constructing effective hedging strategies of European con-

tingent claims in our illiquid financial market model introduced in Chapter 2, Sec-

tion 2.1, the complexity of the results from our case study in Section 2.3 provoke

that explicit characterizations of, e.g., optimal utility-based hedging strategies ap-

pear to be unachievable already in this most elementary illiquid Bachelier setup.

In particular, the additional presence of a random endowment in the utility max-

imization problem will in general rule out optimality of deterministic strategies in

the above framework.

In Chapter 3, we propose a possible remedy. As discussed in Roch and Soner

(2013) and Kallsen and Muhle-Karbe (2014), models of Obizhaeva and Wang (2013)

type with transient price impact are inherently connected to models of Almgren and

Chriss (2001) type with temporary price impact in the sense that the latter can be

regarded as the high-resilience limit of the former when sending the recovery rate of

bid- and ask-prices to infinity. Thus, motivated by the ideas in Kallsen and Muhle-

Karbe (2014), Moreau et al. (2017) or Ekren and Muhle-Karbe (2017), we explain

heuristically how the utility maximization problem with random endowment in

our original illiquid market model from Section 2.1 can be related asymptotically

in a “high resilience” regime to a considerably simpler linear quadratic optimal

hedging problem with quadratic costs induced by linear temporary price impact à

la Almgren and Chriss (2001).

Hedging with temporary price impact: Quadratically optimal

smoothing (Chapter 4)

In the context of illiquid market models, only a few papers directly address the

problem of hedging a European contingent claim in the presence of price impact;

cf., e.g., Rogers and Singh (2010), Almgren and Li (2016), Guéant and Pu (2015),

Naujokat and Westray (2011) or, for a more general framework, Cai et al. (2015).

Motivated by the heuristic considerations from Chapter 3, we relate the hedging

problem in our original illiquid market model from Section 2.1 to a linear quadratic

optimal tracking problem of a frictionless hedging strategy in the presence of tem-

porary price impact as proposed by Almgren and Chriss (2001). We study this

problem in its simplest version in a Bachelier type setup with constant volatility

and constant temporary price impact in Chapter 4. A more general framework will

be considered in Chapter 5.

The papers most closely related to our work mathematically are Rogers and

Singh (2010) and Naujokat and Westray (2011). Rogers and Singh (2010) analyse

the problem of hedging a European contingent claim in a Black-Scholes model in

the presence of purely temporary price impact as in Almgren and Chriss (2001).

Appealing to a quadratic hedging approach, they relate the hedging problem to a

cost optimal tracking problem of the frictionless Black-Scholes delta hedge. Nau-

jokat and Westray (2011) directly study the problem of optimally following a given

target strategy in an illiquid financial market under the same type of liquidity costs.

By contrast to these papers, we will focus on a non-Markovian setup with general

predictable target strategies.

Instead of the more common dynamic programming methods used in the papers

cited above, our approach is convex analytic along the lines of Pontryagin’s max-

imum principle. This allows us to consider general predictable target strategies

and not only continuous diffusion-type processes. This is particularly important

for hedging in illiquid markets when the frictionless reference hedge portfolio pre-

1 Introduction

scribes sizable instantaneous reallocations as, e.g., already in the case of discrete

Asian options which was not covered by the literature so far. We derive first or-

der conditions of the considered quadratic optimization problems which take the

form of a linear forward backward stochastic differential equation (FBSDE). Solu-

tions to these are explicitly available and give us the optimal frictional hedges. In

fact, when considered in a Brownian setting, our approach can be viewed as a spe-

cial case of the stochastic linear quadratic control problem studied by Kohlmann

and Tang (2002); we may also refer to the more general framework investigated

in Chapter 5. Mathematically, our main contribution is the interpretation of the

optimal tracking strategy. Indeed, it turns out that the optimal policy does not

instantaneously trade from its current position towards the current target position

but towards a weighted average of expected future target positions which does not

occur, e.g., in the work of Kohlmann and Tang (2002). An interesting consequence

from a financial point of view is that this averaging allows us to understand how

singularities in the frictionless reference strategy have to be addressed in a model

with frictions: A singularity in the frictionless target hedge is smoothed out when

averaging the weighted future target positions which yields sensible hedging strate-

gies for illiquid markets. Additionally, we also study a constrained version of the

problem where the terminal hedging position is restricted to a certain exogenously

prescribed level. This can be viewed as a way to deal with physical delivery in

derivative contracts. Our explicit solution reveals how the hedger’s focus shifts

systematically from tracking the frictionless target position to attaining the pre-

scribed terminal position. Here, our convex analytic approach allows us to avoid

the consideration of nonlinear Hamilton-Jacobi-Bellmann equations with singular

terminal conditions and the challenges that these entail; cf., e.g., Graewe et al.

(2015) or our discussion in Chapter 5. We also give a sharp characterization of

those terminal positions which can be reached with finite expected trading costs

by characterizing the speed at which the size of these positions is revealed towards

the end.

Conceptually, our result generalizes an observation by Gârleanu and Pedersen

(2013b) who consider mean-variance maximization in homogeneous Markovian

models on an infinite time horizon and interpret their solution as trading towards an

exponentially weighted average of future expected Markowitz portfolios. A similar

interpretation is given by Naujokat and Westray (2011) in their equally Marko-

vian Example 7.1; see also Cartea and Jaimungal (2016) for a similarly Markovian

study of tracking of order flow in high-frequency trading. These strategies as well

as ours contrast with strategies targeting the present frictionless optimum directly,

which are considered in many papers on asymptotically optimal portfolios with

small transaction costs, including Rogers and Singh (2010), Moreau et al. (2017),

Guasoni and Weber (2017), Guasoni and Weber (2015), and Kallsen and Muhle-

Karbe (2014). In all the literature cited above, the authors confine consideration to

diffusion-type target strategies which, at least asymptotically, are equivalent to our

averaged targets. Our approach, by contrast, allows one to deal with general pre-

dictable frictionless target strategies and so the examples considered in this paper

include strategies with jumps or even singularities where the differences between

these hedges become apparent.

Almgren and Li (2016) study a quite similar hedging problem but they consider a

model with permanent price impact which feeds into their target strategies via the

well-known functions for Black-Scholes deltas and gammas. Hence, they consider

a model where the target strategy is also affected by the targeting strategy which

leads to a feedback effect that we are disregarding in our problem formulation. We

refer to the introduction in Rogers and Singh (2010) for a further discussion.

Hedging with stochastic temporary price impact and stochastic

volatility: Linear quadratic control (Chapter 5)

Linear quadratic stochastic optimal control problems (stochastic LQ problems in

short) represent an important class of stochastic control problems and are very well

studied in the literature; cf., e.g., the book by Yong and Zhou (1999), Chapter 6,

for an overview. In fact, the optimal tracking problem considered in Chapter 4is a

prototype of a stochastic LQ problem with linear quadratic cost functional. It is well

known in the literature that the optimal control to such an optimization problem as

well as its optimal value is fully characterized by two coupled backward stochastic

differential equations (BSDEs). A backward stochastic Riccati differential equation

(BSRDE) and, due to the linear component in the objective functional, a linear

BSDE; cf., e.g., Kohlmann and Tang (2002), Section 5.1.

In Chapter 5, we want to extend the results from Chapter 4to a more general

case where we allow for stochastic temporary price impact and stochastic volatility.

In addition, similar to the constrained version of the optimal tracking problem in

Chapter 4, we incorporate a possibly singular stochastic terminal state constraint.

That is, in case of a possible but not necessarily almost sure occurrence of specific

market conditions, the agent may require to drive her hedging portfolio imperatively

towards a predetermined random value at terminal time.

Mathematically, it is less obvious how to tackle this delicate singularity and

how to compute the optimal control as well as the optimal value. The underlying

involved BS(R)DEs from the unconstrained case will both now exhibit with positive

probability a singularity at final time in this case. The solvability of the possibly

singular BSRDE has been recently studied in Kruse and Popier (2016a), Popier

(2016) and Graewe et al. (2015) and we will assume the solution process to be given.

In contrast, the singularity in the terminal condition of the linear BSDE is rather

unpleasant because it also involves the desired final target position which leads to

an ill-posed problem. As a consequence, one needs to find a suitable substitute for

the linear BSDE. Moreover, it is not clear a priori whether the stochastic optimal

tracking problem with terminal state constraint admits an optimal finite solution

1 Introduction

at all. This has to be precluded via identifying appropriate conditions.

Having at hand the results from Chapter 4in case of constant temporary price

impact and constant volatility, we will formulate and solve a suitable variant of

the considered optimal tracking problem with terminal state constraint. In fact, in

the present more general setting of Chapter 5, given the solvability of the singular

BSRDE, it turns out that similar to the obtained solutions in Chapter 4, a key role

is played by a generalized version of the above emerged optimal signal process. In

particular, this process will serve as an adequate substitute for the linear BSDE.

As we will see, it also allows similarly to the representation in Chapter 4for an

intuitively appealing interpretation of the optimal control and makes transparent

the associated optimal costs. The results from Chapter 5can therefore be regarded

as the natural general extensions of the results from Chapter 4.

This signal process, together with the solution process of the singular BSRDE,

will provide the main tool not only in solving but also in tackling the considered

LQ problem with its delicate stochastic terminal state constraint. Our main idea is

to resolve the technical difficulties due to the state constraint by moving away in a

suitable manner from terminal time and to approximate the LQ problem via a con-

sistent truncation in time. Specifically, we propose to consider the original problem

as a limit of “tame” stochastic LQ problems with a strictly shorter time horizon

at which we impose a terminal penalization term with “classical” finite coefficient.

The optimal signal process will be the proper key ingredient for choosing these

penalizations in a time consistent manner. It turns out that the optimal controls

and the corresponding costs to these time truncated LQ problems can be identified

and prolonged without difficulties to the original terminal time where the desired

stochastic state constraint is satisfied. As a byproduct, we obtain necessary and

sufficient conditions in terms of the optimal signal process and the solution process

of the singular BSRDE under which the LQ problem with stochastic terminal state

constraint actually admits a finite optimal value.

Stochastic linear quadratic control problems with almost sure and deterministic

terminal state constraint, i.e., targeting the terminal position of zero, are exten-

sively studied in the optimal liquidation literature; cf., e.g., Schied (2013), Ankirch-

ner et al. (2014), Graewe et al. (2015), Graewe and Horst (2016) and Graewe et al.

(2017). Kruse and Popier (2016a) consider optimal liquidation at terminal time

with positive but not necessarily almost sure probability. Ankirchner and Kruse

(2015) incorporate a specific non-zero stochastic terminal state constraint where

the random target position is gradually revealed up to terminal time. A general

class of stochastic control problems including LQ problems with terminal states

being constrained to a convex set were studied by Ji and Zhou (2006). However, to

the best of our knowledge, stochastic optimal tracking problems with random tar-

get position and not almost sure occurring terminal state constraint as considered

in Chapter 5have not yet been investigated.

2 Optimal Investment with Transient Price Impact

In this chapter we propose a variant of the idealized limit order book model by

Obizhaeva and Wang (2013) which allows for both selling and buying stock. The

coupled bid- and ask-price dynamics of our price impact model induce convex liq-

uidity costs on the trading strategies. We provide existence of an optimal solution

to the classical Merton problem of maximizing expected utility from terminal liqui-

dation wealth at a finite planning horizon in Section 2.2. In the specific case when

market uncertainty is generated by an arithmetic Brownian motion with drift and

the investor exhibits constant absolute risk aversion, we show that the resulting

singular optimal control problem reduces to a deterministic optimal tracking prob-

lem of the frictionless constant Merton portfolio in the presence of convex liquidity

costs. We construct the solution explicitly by methods from convex analysis and

calculus of variations in Section 2.3. As expected by previous studies in the liter-

ature, it is optimal to trade towards the frictionless Merton position, taking into

account the initial bid-ask spread as well as the optimal liquidation of the accrued

position when approaching terminal time. It turns out that this leads to a surpris-

ingly rich phenomenology of possible trajectories for the optimal share holdings; cf.

the synopsis in Section 2.3.5. The technical proofs are deferred to Section 2.4.

2.1 A price impact model

We fix a filtered probability space (Ω,F,(Ft)t≥0,P)satisfying the usual conditions

of right continuity and completeness and consider an investor whose trades in a

risky asset affect its market prices in an adverse manner. For our specification of

her price impact, we propose a variant of the block-shaped limit order book model

introduced by Obizhaeva and Wang (2013).

Specifically, the investor’s trading strategy is described by a pair X= (X↑, X↓)

of predictable, nondecreasing, right-continuous processes where X↑= (X↑

t)t≥0,

X↓= (X↓

t)t≥0denote, respectively, the cumulative purchases and sales of the risky

asset until time t≥0with X↑

0−≜X↓

0−≜0. Trading takes place via market orders

in a block-shaped limit order book at the best bid- and ask-prices BXand AX

which are specified as the solution to the following coupled system of controlled

diffusions

dAX

t=dPt+ηdX↑

t−1

2κ(AX

t−−BX

t−)dt

dBX

t=dPt−ηdX↓

t+1

2κ(AX

t−−BX

t−)dt

(t≥0) (2.1)

2 Optimal Investment with Transient Price Impact

with given parameters η > 0,κ > 0,AX

0−≜A0>0and BX

0−≜B0>0. The

interpretation of the bid- and ask-price dynamics in (2.1) is the following: Both

processes AXand BXare driven by some common exogenous fundamental random

shock dPtmodeled by a continuous semimartingale (Pt)t≥0with initial value P0−≜

(A0+B0)/2. The process (Pt)t≥0can also be regarded as the unaffected price

process. Due to finite market depth 1/η ∈(0,∞)which can be interpreted as the

height of a block-shaped limit order book, a buy order dX↑

tincurs an impact and

increases the best ask-price AXby the amount ηdX↑

twhereas the best bid-price

BXis not directly affected. After completion of each buy trade, ask- and bid-

prices revert to each other at some finite resilience rate κ > 0. The effects of sell

orders dX↓

ton the best bid-price BXin (2.1) are analogous. Put differently, price

impact is transient and does not vanish instantaneously but persists and decays over

time at the exponential rate κ. We will assume for simplicity that both illiquidity

parameters, the instantaneous price impact factor ηas well as the resilience rate κ,

are constant. According to the bid- and ask-price dynamics in (2.1), the evolution

of the bid-ask spread ζX

t≜AX

t−BX

tis given by

dζX

t=η(dX↑

t+dX↓

t)−κζX

t−dt (t≥0) (2.2)

with initial value ζX

0−≜ζ0≥0and right-continuous solution

ζX

t=e−κ(t−s)(ζX

s−+η∫[s,t]

eκ(u−s)(dX↑

u+dX↓

u))(0 ≤s≤t).(2.3)

Let us now derive the investor’s wealth dynamics corresponding to a trading

strategy X= (X↑, X↓). First, we associate to Xthe self-financing portfolio process

(ξX

t, φX

t)t≥0with some given initial values (ξX

0−, φX

0−)∈R2where ξX

tdenotes the

amount of cash and φX

t≜φX

0−+X↑

t−X↓

tthe number of shares of the risky asset held

at time t≥0. Assuming zero interest rates, the self-financing condition dictates

that the cash balance ξXchanges only due to trading activity X, that is, we have

dξX

t=−(AX

t−+1

2η∆X↑

t)dX↑

t+(BX

t−−1

2η∆X↓

t)dX↓

t(t≥0)

with ∆X↑,↓

t≜X↑,↓

t−X↑,↓

t−, respectively. Observe that the actual execution price to,

e.g., buy dX↑

tshares at time tis given by AX

t−+η∆X↑

t/2where η∆X↑

t/2accounts

for the impact a non-infinitesimal order incurs; cf., e.g., also Alfonsi et al. (2010) or

Predoiu et al. (2011). Analogous for sell orders. The investor’s total wealth at any

time t≥0is now expressed in terms of the liquidation value Vt(X)of her current

portfolio (ξX

t, φX

t)which we define as follows.

Definition 2.1.1 (Liquidation wealth).The investor’s liquidation wealth process

(Vt(X))t≥0associated to her portfolio process (ξX, φX)with trading strategy X=

(X↑, X↓)and initial endowment (ξX

0−, φX

0−)∈R2is defined as

Vt(X)≜ξX

t+1

2(AX

t+BX

t)φX

t−(1

2ζX

t|φX

t|+1

2η(φX

t)2)(t≥0) (2.4)

2.1 A price impact model

with initial value V0−(X)≜ξX

0−+φX

0−(A0+B0)/2−(ζ0|φX

0−|+η(φX

0−)2)/2.

Note that the liquidation value Vt(X)in (2.4) decomposes into two parts: The

first part represents the portfolio’s book value ξX

t+φX

t(AX

t+BX

t)/2where the value

of the position φX

tin the risky asset is measured in terms of the mid-quote price

(AX

t+BX

t)/2. The second part ζX

t|φX

t|/2+η(φX

t)2/2represents the corresponding

liquidation costs which are incurred by the bid-ask spread ζX

tas well as the instan-

taneous price impact ηwhen unwinding in one single block trade the φX

tshares.

The following lemma shows that the dynamics of the liquidation wealth process

(Vt(X))t≥0in (2.4) conveniently separate into the common frictionless wealth and

a nonnegative, convex cost functional.

Lemma 2.1.2. The liquidation wealth process (Vt(X))t≥0of a strategy X= (X↑, X↓)

defined in (2.4)allows for the decomposition

Vt(X) = V0−(X) + L0−(X) + ∫t

φX

sdPs−Lt(X) (t≥0) (2.5)

where (Lt(X))t≥0denotes the liquidity costs defined as

Lt(X)≜1

4η(η|φX

t|+ (ζX

t−e−κtζ0))2+1

2|φX

t|e−κtζ0+η

4(φX

0−)2

2∫[0,t]

e−κsζ0(dX↑

s+dX↓

s) + κ

2η∫t

(ζX

s−−e−κsζ0)2ds (2.6)

with initial value L0−(X)≜ζ0|φX

0−|/2 + η(φX

0−)2/2. In particular, for all t≥0the

functional Lt(X)is convex in Xand satisfies

Lt(X)≥η

4e−2κt(X↑

t+X↓

t)2+κη

2∫t

e−2κs(X↑

s+X↓

s)2ds ≥0.(2.7)

By definition, the quantity V0−(X)+L0−(X) = ξX

0−+φX

0−P0−in (2.5) represents

the initial wealth’s book value, or initial frictionless wealth, of strategy Xwith

initial endowment (ξX

0−, φX

0−).

Remark 2.1.3.1. Compared to other price impact models which are used in

the literature in the context of studying optimal investment problems, our

price impact in (2.1) depends on the trading volume of the investor in the

spirit of Obizhaeva and Wang (2013) and not on the trading rate as, e.g.,

in Gârleanu and Pedersen (2013a,b) or Forde et al. (2016). They adapt

purely temporary price impact as proposed by Almgren and Chriss (2001).

In Guasoni and Weber (2017,2015,2016), temporary price impact is not

only induced by the trading rate but also depends on the investor’s total

wealth. Our model captures transient price impact which persists and decays

over time. As a consequence, trading strategies X= (X↑, X↓)are no longer

restricted to be absolutely continuous but also comprise block trades. In

2 Optimal Investment with Transient Price Impact

fact, our modelling approach is similar to the one proposed in Roch and

Soner (2013) which is adopted in a reduced form with zero spread by Soner

and Vukelja (2016) who study an utility maximization problem. They allow

for more general stochastic dynamics for the market depth and the resilience

rate. Another difference is that our bid- and ask-prices in (2.1) revert to each

other and not to some auxiliary reference price as in Roch and Soner (2013).

2. Recall that proportional transaction costs as considered, e.g., in Davis and

Norman (1990), are linear in the risky asset holdings. Purely temporary price

impact which is linear in the trading rate of absolutely continuous investment

strategies as considered in Gârleanu and Pedersen (2013a,b) or Guasoni and

Weber (2017,2016) induces quadratic liquidity costs on the latter. Forde

et al. (2016) and Guasoni and Weber (2015) allow for nonlinear price impact

which introduces a dependence of the incurred trading costs on a fractional

power of the turnover rates. In our model above, price impact in (2.1) is still

linear in the trading strategy X= (X↑, X↓)but the induced liquidity costs

in (2.6) are convex rather than purely quadratic because of the emergence

of the absolute value function in (2.6). Nonetheless, as we will show below,

convexity comes along with a pleasant tractability.

2.2 Optimal investment problem

Now, let us consider an investor who aims to trade optimally in the price im-

pact model introduced in the preceding Section 2.1. The investor’s preferences are

described by a utility function u:R→Rin C1(R)which is strictly concave, in-

creasing and bounded from above. She wants to maximize expected utility from

her terminal liquidation wealth VT(X)at some finite planning horizon T > 0as

defined in (2.4) by following a trading strategy X= (X↑, X↓)with given initial

endowment ξX

0−≜ξ0∈Rin cash and φX

0−≜φ0∈Rshares of the risky asset.

Her corresponding initial wealth and the associated liquidation costs are denoted

by V0≜V0−(X)and L0≜L0−(X)with given initial bid-ask spread ζX

0−=ζ0≥0.

In other words, in view of Lemma 2.1.2 above, the agent’s aim is to solve the

optimization problem

Eu(VT(X)) = Eu(V0+L0+∫T

φX

tdPt−LT(X))→max

X=(X↑,X↓)∈X(2.8)

over all admissible trading policies

X≜{(Xt)t≥0= (X↑

t, X↓

t)t≥0:X↑, X↓right-continuous,

predictable, nondecreasing processes with X↑

0−≜X↓

0−≜0}.

2.2 Optimal investment problem

The main tool allowing us to provide existence of an optimal strategy to the

maximization problem in (2.8) is given by the following convex compactness result

for processes of finite variation.

Lemma 2.2.1 (Guasoni (2002), Lemma 3.4).Consider a sequence of strategies

(Xn)n≥1⊂Xsuch that conv({X↑,n

T+X↓,n

T:n≥1})is bounded in L0(Ω,F,P).

Then there exists a strategy X∈Xand a sequence (˜

Xn)n≥1⊂Xof cofinal

convex combinations, i.e., ˜

Xn∈conv(Xn, Xn+1, . . .)for all n≥1, converging to

Xpointwise:

lim

n→∞

X↑,↓,n

t(ω) = X↑,↓

t(ω)for all t∈[0, T], ω ∈Ω.(2.9)

Proof. Lemma 3.4 in Guasoni (2002) yields (˜

Xn)n≥1as described above as well as

that for almost every ω∈Ωwe have the convergence limn→∞ ˜

X↑,↓,n

t(ω) = X↑,↓

t(ω)

everywhere on [0, T], except, possibly, in the discontinuity points of the limit pro-

cess X↑,↓

·(ω). In addition, by the same reasoning as in the proof of Proposition 3.4

in Campi and Schachermayer (2006), namely by appealing to a section theorem

(Théorème 117 in Dellacherie and Meyer (1975)) as well as Cantor’s diagonal argu-

ment, the convergence in (2.9) can also be established at the discontinuity points

of X↑,↓

·(ω)∈Xand thus for every t∈[0, T]. Finally, we can assume without loss

of generality that the convergence in (2.9) also holds for every ω∈Ω.

Another important ingredient is provided by the upper semi-continuity of the

liquidation wealth VT(X)in X∈Xgiven in (2.5).

Lemma 2.2.2. Let T > 0and let (Xn)n≥1⊂Xbe a sequence of strategies with

same initial endowment (ξX

0−, φX

0−) = (ξ0, φ0)such that Xn→X∈Xpointwise

on Ω×[0, T]. Then, it holds that

lim sup

n→∞ VT(Xn)≤VT(X)pointwise on Ω.

As a consequence, due to convexity of the liquidity cost functional LT(X)in

X∈Xby virtue of Lemma 2.1.2, we can establish the following existence and

uniqueness result for the optimization problem in (2.8).

Theorem 2.2.3. There exists a unique strategy ˆ

X= ( ˆ

X↑,ˆ

X↓)∈Xsuch that

Eu(VT(ˆ

X)) ≥Eu(VT(X)) for all strategies X= (X↑, X↓)∈X.

Proof. Consider a maximizing sequence (Xn)n≥1⊂Xsuch that

u∗≜sup

X∈X

Eu(VT(X)) = lim

n→∞Eu(VT(Xn)) ∈(−∞, u(∞)).

We can assume without loss of generality that the sequence (Xn)n≥1belongs to the

level-set L0:= {X∈X:Eu(VT(X)) ≥Eu(VT(0)) = u(V0+L0)}. Moreover, due

2 Optimal Investment with Transient Price Impact

to Lemma 2.4.1 below, it holds that conv({X↑

T+X↓

T:X∈L0})is L0(Ω,F,P)-

bounded. Hence, by virtue of the compactness result in Lemma 2.2.1, there exists

a strategy ˆ

X∈Xand a sequence (˜

Xn)n≥1⊂Xof convex combinations ˜

Xn∈

conv(Xn, Xn+1, . . .)such that ˜

Xn→ˆ

Xpointwise for n↑ ∞. We claim that ˆ

Xis

the optimal solution to problem (2.8). Indeed, since the liquidity costs are convex,

(˜

Xn)n≥1is again a maximizing sequence. Specifically, given a finite number of

strictly positive weights (λn

m)m≥nof ˜

Xn, we have

u(VT(˜

Xn)) ≥∑

m≥n

λn

mu(VT(Xm))

where we also used monotonicity and concavity of u. Taking expectations and pass-

ing to the limit in the above inequality yields limn→∞ Eu(VT(˜

Xn)) ≥u∗. Moreover,

by upper-semicontinuity of the liquidation wealth provided in Lemma 2.2.2 and Fa-

tou’s Lemma we obtain

u∗≥Eu(VT(ˆ

X)) ≥Elim sup

n→∞ u(VT(˜

Xn)) ≥lim sup

n→∞

Eu(VT(˜

Xn)) ≥u∗.

Uniqueness of the optimizer ˆ

Xfollows from strict concavity of the utility function

uand again convexity of the liquidity costs.

Remark 2.2.4.

1. Note that absence of arbitrage opportunities for the unaffected price process

(Pt)t≥0in (2.1) does not play any role for the existence of an optimal invest-

ment strategy for the utility maximization problem in (2.8). In particular,

the no-arbitrage condition is not as crucial in our setup as in frictionless mar-

ket models because arbitrage opportunities are not scalable in our nonlinear

model exhibiting nonnegative, convex liquidity costs in (2.6). Therefore, we

also do not have to impose further admissibility constraints on the set Xof

allowed policies X= (X↑, X↓).

2. Observe that the existence proof of Theorem 2.2.3 above also works in the

presence of, e.g., a bounded initial random endowment H∈FTin the utility

maximization problem in (2.8).

2.3 Case study: Illiquid Bachelier model with exponential utility

In this section, we will investigate the utility maximization problem from terminal

liquidation wealth as formulated in (2.8) in the specific case when market uncer-

tainty dPtin our price impact model in (2.1) above is generated by a Brownian

motion with drift µ > 0and volatility σ > 0. In other words, we assume that the

unaffected price process (Pt)t≥0satisfies

dPt=µdt +σdWt(t≥0) (2.10)

2.3 Case study: Illiquid Bachelier model with exponential utility

where (Wt)t≥0denotes a standard Brownian motion on the given filtered probability

space (Ω,F,(Ft)t≥0,P). In addition, we assume that the inverstor’s preferences

are described by an exponential utility function

u(x) = −e−αx (x∈R)

with constant absolute risk aversion parameter α > 0. For simplicity, let us further

suppose that the investor is confined to investment strategies X= (X↑, X↓)∈X

which are bounded. Within this setup, the optimization problem in (2.8) now reads

E[−exp{−α(µ∫T

φX

tdt +σ∫T

φX

tdWt−LT(X))}]→max

X=(X↑,X↓)∈¯

(2.11)

with ¯

X≜{X∈X: esssup(X↑

t+X↓

t)<∞for all t≥0P-a.s.}. Note that due

to the scaling property of the exponential utility function, the optimal strategy

in (2.11) does not depend on the investor’s initial frictionless wealth V0+L0.

Remark 2.3.1.Our restriction to the set ¯

Xis due to technical reasons which become

more apparent in the proof of Proposition 2.3.2 below. Specifically, we require that

the stochastic exponential E(−ασ ∫T

0φX

tdWt)is a true martingale for all considered

strategies Xwhich is clearly the case if X∈¯

X. We conjecture that this martingale

property actually holds true for all X∈Xsuch that E[u(VT(X))] >−∞. However,

in the context of our present case study we do not further pursue the analysis in

this direction.

It is well known in the literature that in the frictionless case with η=ζ0= 0,

i.e., AX=BX=Pin (2.1) and LT(X) = 0 in (2.6) for any X∈X, the optimal

strategy ˆ

X0= ( ˆ

X0,↑,ˆ

X0,↓)to problem (2.11) is simply a deterministic buy-and-

hold-strategy given by

dˆ

X0,↑

t=µ

ασ2δ0(dt)and dˆ

X0,↓

t=µ

ασ2δT(dt)on [0, T]

where δtdenotes the Dirac measure in t. That is, the optimal frictionless share

holdings φ0in the risky asset are constant over time and given by the Merton

portfolio

φ0

t≜µ

ασ2(0 ≤t≤T)(2.12)

which is acquired at time 0and unwound at time Twith, respectively, an initial and

final block trade. Of course, the main question is what will change when accounting

for illiquidity, i.e., price impact induced by finite market depth and market tightness

given by the bid-ask spread. It is intuitively sensible to expect that the optimal

frictional portfolio for problem (2.11) will trade towards the desired Merton position

in (2.12). In fact, in the presence of price impact η > 0it turns out that it is still

enough to study problem (2.11) for deterministic strategies which translates into

an optimal tracking problem of the frictionless optimal portfolio position φ0.

2 Optimal Investment with Transient Price Impact

Proposition 2.3.2. For given time horizon T > 0, initial position φ0∈Rand

initial spread ζ0≥0, the optimal investment strategy of the maximization prob-

lem in (2.11)is deterministic and coincides with the minimizer of the convex cost

functional

JT(X)≜LT(X) + ασ2

2∫T

0(φX

t−µ

ασ2)2

dt →min

X=(X↑,X↓)∈Xd(2.13)

with Xd≜{X∈X:X= (X↑, X↓)deterministic}.

Remark 2.3.3.Let us point out that the general existence and uniqueness result

from Theorem 2.2.3 above is not directly applicable to the optimization problem

in (2.11) and thus (2.13) because of the restricted set ¯

X⊂Xof allowed strategies.

However, observe that by the same reasoning as in the proof of Theorem 2.2.3 exis-

tence and uniqueness of a deterministic optimizer in Xd⊂¯

Xto the deterministic

tracking problem in (2.13) for any time horizon T > 0, initial position φ0∈Rin

the risky asset and any initial bid-ask spread ζ0≥0can be guaranteed.

Proof of Proposition 2.3.2.We follow the same idea as in Schied et al. (2010).

For convenience, let us define the cost functional

JT(X)≜LT(X)−µ∫T

φX

tdt +ασ2

2∫T

(φX

t)2dt =JT(X)−µ2

2ασ2T

for all X∈¯

Xand let us set ˜

J∗

T≜infX∈Xd˜

JT(X). Next, let X∈¯

Xbe an

arbitrary strategy and denote by

dPX

≜E(−ασ ∫T

φX

tdWt)= exp(−ασ ∫T

φX

tdWt−α2σ2

2∫T

(φX

t)2dt)

the induced probability measure on (Ω,FT,P). Then, it follows by a change of

measure that

E[u(VT(X))] = E[−exp(−α∫T

φX

tdPt+αLT(X))]

=EPX[−exp(αLT(X)−αµ ∫T

φX

tdt +α2σ2

2∫T

(φX

t)2dt)]

=EPX[−eα˜

JT(X)]≤ −eα˜

J∗

(2.14)

with equality holding true for the unique deterministic minimizer X∈Xdof ˜

JT;

cf. also Remark 2.3.3. Thus, the maximizer of the right-hand side in (2.14) over

all admissible strategies in ¯

Xwhich corresponds to our original problem in (2.11)

is actually given by the deterministic strategy attaining the value ˜

J∗

The minimization problem in (2.13) can be regarded as a deterministic optimal

tracking problem of the frictionless Merton portfolio φ0≡µ/(ασ2)in the presence

2.3 Case study: Illiquid Bachelier model with exponential utility

of trading costs measured by LT(·). The optimal strategy ˆ

Xseeks to minimize

both the squared deviation of its share holdings φˆ

Xfrom the preferred constant

position φ0as well as the incurred liquidity costs LT(ˆ

X)which are levied on its

trading activity ˆ

X= ( ˆ

X↑,ˆ

X↓)due to market tightness and the finite market’s

depth. Moreover, liquidation is costly and the optimizer additionally has to take

into account unwinding his accrued position in the risky asset in an optimal manner

when approaching terminal time T. Consequently, the following questions arise

naturally: Given an arbitrary initial position φ0∈Rin the risky asset, how is

the desired position µ/(ασ2)approached? How is it unwound in the end? What

happens if the initial spread ζ0is very large and/or the trading period T > 0is

very short? In fact, we will answer these questions and characterize explicitly the

optimal solution to the minimization problem in (2.13) in the subsequent Sections

2.3.1 to 2.3.5 below.

Remark 2.3.4.1. For deterministic strategies X∈Xdthe liquidation wealth

VT(X)in (2.5) in the present illiquid Bachelier model with unaffected price

process (2.10) is normally distributed. Therefore, the maximization problem

in (2.11) and thus the minimization problem in (2.13) is equivalent to the

problem of maximizing a mean-variance criterion given by

E[VT(X)] −α

2var(VT(X))

=V0+L0+µ∫T

φX

tdt −LT(X)−ασ2

2∫T

(φX

t)2dt;

cf. also the discussion in Schied et al. (2010).

2. Let us mention that the deterministic optimal tracking problem in (2.13) is si-

miliar to the stochastic tracking problem studied in Chapter 4below. Therein,

we will investigate the problem of minimizing the L2(P⊗dt)-distance of a port-

folio process φXfrom a predictable stochastic target process (ξt)0≤t≤Tin the

presence of temporary price impact as in Almgren and Chriss (2001). The

process (ξt)0≤t≤Trepresents, e.g., an optimal investment or hedging strategy

adopted from a frictionless setting. Specifically, investment strategies φX

are restricted to be absolutely continuous and quadratic costs are levied on

the respective trading rates ˙φX. In contrast, the liquidity costs LT(·)in the

present setup are induced by market tightness and transient price impact à

la Obizhaeva and Wang (2013). A heuristic connection between these two

problems will be provided in Chapter 3.

3. A similar stochastic singular control problem is studied in Horst and Naujokat

(2014). The authors derive a suitable version of the stochastic maximum

principle and characterize optimal strategies in terms of forward-backward

stochastic differential equations. For our deterministic optimization problem

in (2.13), we will use a convex analytic approach.

2 Optimal Investment with Transient Price Impact

2.3.1 First order optimality conditions

Since the objective functional JT(·)of the minimization problem in Proposition

2.3.2 is convex, we can use tools from convex analysis and calculus of variations in

order to provide a characterization of the optimal solution in terms of necessary and

sufficient first order conditions. Specifically, let us note that the convex functional

JT(·)is supported on Xdby the infinite-dimensional buy- and sell-subgradients

defined as

ϱ∇↑

tJT(X)≜∫T

t(κe−κ(u−t)ζX

u+ασ2(φX

u−µ

ασ2))du

2(η|φX

T|+ζX

T)e−κ(T−t)(2.15)

+η

2φX

T+1

2signϱ(φX

T)ζX

T(0 ≤t≤T)

and

ϱ∇↓

tJT(X)≜∫T

t(κe−κ(u−t)ζX

u+ασ2(µ

ασ2−φX

u))du

2(η|φX

T|+ζX

T)e−κ(T−t)(2.16)

−η

2φX

T−1

2signϱ(φX

T)ζX

T(0 ≤t≤T)

in the sense of Lemma 2.3.6 below.

Remark 2.3.5.The map x↦→ signϱ(x)appearing in the definition of the buy-

and sell-subgradients in (2.15) and (2.16) above represents the subgradient of the

absolute value function x↦→ |x|(cf. proof of Lemma 2.3.6 in Section 2.4 below)

and therefore allows for an arbitrary value signϱ(0) ≜ϱ∈[−1,1] when φX

T= 0. In

this case the subgradients are actually set-valued. The dependence on the value ϱ

is indicated by the left-hand superscript in the operator symbols ϱ∇↑and ϱ∇↓. To

alleviate the notation, we will simply write ∇↑,∇↓and sign(·)most of the time

unless a specification of the value ϱbecomes indispensable.

Lemma 2.3.6. For any two strategies X, Y ∈Xdwith the same initial position

φY

0−=φX

0−and initial spread ζ0≥0we have

JT(Y)−JT(X)≥∫[0,T ]

ϱ∇↑

tJT(X)(dY ↑

t−dX↑

t) + ∫[0,T ]

ϱ∇↓

tJT(X)(dY ↓

t−dX↓

with ϱ∇↑JT(X)and ϱ∇↓JT(X)as defined in (2.15)and (2.16), respectively.

For any nondecreasing, right-continuous process Zwith Z0−≜0, let us further

define the set

{dZ > 0}≜{t∈[0, T] : Zt−< Zufor all u > t}(2.17)

2.3 Case study: Illiquid Bachelier model with exponential utility

and observe that for any continuous G= (Gt)0≤t≤Twe have

∫T

GtdZt=∫{dZ>0}

GtdZt.

Having this notation at hand, we can now state necessary and sufficient first order

optimality conditions for the minimization problem stated in Proposition 2.3.2.

Proposition 2.3.7 (First order condition).The strategy ˆ

X= ( ˆ

X↑,ˆ

X↓)in Xd

solves the optimization problem in (2.13)if and only if the following conditions

hold true:

(i) ∇↑

tJT(ˆ

X)≥0for all t∈[0, T]with ‘=’ on the set {dˆ

X↑>0},

(ii) ∇↓

tJT(ˆ

X)≥0for all t∈[0, T]with ‘=’ on the set {dˆ

X↓>0}.

In case φˆ

T= 0, the conditions in (i) and (ii) are meant to hold for ϱ∇↑and ϱ∇↓

with some suitable value ϱ∈[−1,1].

Proof. First, assume that ˆ

X= ( ˆ

X↑,ˆ

X↓)satisfies conditions (i) and (ii) (for some

suitable ϱ∈[−1,1] in case φˆ

T= 0) and let Y∈Xdbe an arbitrary competing

strategy with same initial endowment φY

0−=φˆ

0−. Then, by virtue of Lemma 2.3.6

above, it holds that

JT(Y)−JT(ˆ

X)≥∫[0,T ]

ϱ∇↑

tJT(ˆ

X)dY ↑

t+∫[0,T ]

ϱ∇↓

tJT(ˆ

X)dY ↓

−∫[0,T ]

ϱ∇↑

tJT(ˆ

X)dˆ

X↑

t−∫[0,T ]

ϱ∇↓

tJT(ˆ

X)dˆ

X↓

t≥0

which implies JT(Y)≥JT(ˆ

X). Necessity of the conditions (i) and (ii) follows from

the uniqueness of the optimizer and its explicit construction as discussed below in

Sections 2.3.3 and 2.3.4.

In view of Lemma 2.3.6, the quantities ϱ∇↑

tJT(X)and ϱ∇↓

tJT(X)in (2.15) and

(2.16) can be regarded as (lower bounds for) the marginal costs which are incurred

by an additional infinitesimal buy order and sell order at time t, respectively, other-

wise following strategy X. Hence, according to the first order conditions in Propo-

sition 2.3.7, the optimal strategy ˆ

Xacts so as to keep these additional marginal

costs from intervention always nonnegative and only intervenes, i.e., buys or sells

the risky asset, when the corresponding marginal costs ϱ∇↑

tJT(ˆ

X)or ϱ∇↓

tJT(ˆ

vanish. In this regard, observe that, loosely speaking, the subgradients in (2.15)

and (2.16) can be interpreted as capturing the interaction between the future devia-

tion of the controlled share holdings φXfrom the target µ/(ασ2), the corresponding

incurred spread ζXas well as their final values φX

Tand ζX

T. Due to market tight-

ness, it is intuitively sensible to expect that it will never be optimal to purchase

and sell the risky asset at the same time. In fact, this holds true in our setting and

is a direct consequence of the structure of the subgradients.

2 Optimal Investment with Transient Price Impact

Lemma 2.3.8. Let X= (X↑, X↓)∈Xd,X= (0,0), be a strategy such that

∇↑

tJT(X) = 0 for some t∈[0, T]. Then, ∇↓

tJT(X)>0. The same holds true when

interchanging ↑with ↓.

Remark 2.3.9 (Dynamic programming principle).Note that for any strategy X=

(X↑, X↓)∈Xdthe subgradients of JT(·)in (2.15) and (2.16) at time t∈[0, T]

only depend on the values φX

t−,X↑

t−,X↓

t−,ζX

t−, the remaining time to maturity

T−tand the future evolution of the strategy (Xu)t≤u≤T. This property, together

with the uniqueness of the optimal solution to problem (2.13), cf. Remark 2.3.3,

implies that the dynamic programming principle (or so-called Bellman optimality)

is applicable in our setting. Specifically, let ˆ

X∈Xddenote the unique optimal

strategy for problem (2.13) with time horizon T > 0, initial position φˆ

0−=φ∈R

and initial spread ζˆ

0−=ζ≥0. Henceforth, we will use the notation ˆ

XT,ζ,φ =

(ˆ

XT,ζ,φ,↑,ˆ

XT,ζ,φ,↓)in order to emphasize the dependence of the optimal control on

the problem data (T, ζ, φ). Then, for any 0≤t≤Tit follows that the strategy

XT−t,ζ ˆ

t−,φ ˆ

t−

s≜ˆ

XT,ζ,φ

t+s−ˆ

XT,ζ,φ

t−(0 ≤s≤T−t)(2.18)

with problem data (T−t, ζ ˆ

t−, φ ˆ

t−)obviously belongs to Xdand is optimal for

problem (2.13) with time horizon T−t > 0, initial position φˆ

t−∈Rand initial

spread ζˆ

t−≥0. Indeed, it holds that

∇↑,↓

sJT−t(ˆ

XT−t,ζ ˆ

t−,φ ˆ

t−) = ∇↑,↓

t+sJT(ˆ

XT,ζ,φ) (0 ≤s≤T−t)

which can be regarded as a flow-property of the subgradients. This will allow

us to identify optimal controls via a “backward induction in time” and to “glue

together” optimal solutions of problem (2.13) for different maturities. This will be

our methodology for the solution to our optimal tracking problem.

2.3.2 The state space

Our objective is to solve the optimization problem formulated in (2.13) for any given

problem data (T, ζ0, φ0). For this purpose, let us introduce the three-dimensional

state space

S≜{(τ, ζ, φ) : τ≥0, ζ ≥0, φ ∈R} ⊂ R3(2.19)

with time to maturity τ, spread ζand number of shares φof the risky asset. For

any triplet or problem data (τ, ζ, φ)in the state space Swe want to identify the

corresponding unique optimal strategy ˆ

Xτ,ζ,φ which minimizes the functional Jτ(·)

in (2.13) with time horizon τ, initial share holdings φˆ

Xτ,ζ,φ

0−=φand initial spread

value ζˆ

Xτ,ζ,φ

0−=ζ(cf. Remark 2.3.16 below for our convention in the special case

τ= 0). Specifically, we want to describe the evolution of the optimally controlled

system

(τ−t, ζ ˆ

Xτ,ζ,φ

t, φ ˆ

Xτ,ζ,φ

t)0≤t≤τ⊂S

2.3 Case study: Illiquid Bachelier model with exponential utility

in the state space. Intuitively, the first order optimality conditions formulated

in Proposition 2.3.7 suggest a separation of the state space Sinto two action

regions, i.e., a buying- and a selling-region, as well as a non-action or waiting-

region for the optimizer ˆ

Xτ,ζ,φ. Depending on whether the optimally controlled

triplet (τ−t, ζ ˆ

Xτ,ζ,φ

t, φ ˆ

Xτ,ζ,φ

t)at time t∈[0, τ]is located in the buying-, selling-

or waiting-region, the corresponding optimal strategy ˆ

Xτ,ζ,φ buys, sells or does

nothing, respectively, at this time instant t. Hence, Proposition 2.3.7, Lemma 2.3.8

as well as Remark 2.3.9 motivate the following definition of the buying-, selling-

and waiting-region.

Definition 2.3.10 (Buying-, selling-, waiting-region).

1. We define the buying-region as

Rbuy ≜{(τ, ζ, φ)∈S:the optimal strategy ˆ

Xτ,ζ,φ ∈Xdsatisfies

ϱ∇↑

0Jτ(ˆ

Xτ,ζ,φ) = 0 for some ϱ

and ˆ

Xτ,ζ,φ,↑

0>0}

(2.20)

and the boundary of the buying-region as

∂Rbuy ≜{(τ, ζ, φ)∈S:the optimal strategy ˆ

Xτ,ζ,φ ∈Xdsatisfies

ϱ∇↑

0Jτ(ˆ

Xτ,ζ,φ) = 0 for some ϱ

and ˆ

Xτ,ζ,φ,↑

0= 0}.

(2.21)

2. We define the selling-region as

Rsell ≜{(τ, ζ, φ)∈S:the optimal strategy ˆ

Xτ,ζ,φ ∈Xdsatisfies

ϱ∇↓

0Jτ(ˆ

Xτ,ζ,φ) = 0 for some ϱ

and ˆ

Xτ,ζ,φ,↓

0>0}

(2.22)

and the boundary of the selling-region as

∂Rsell ≜{(τ, ζ, φ)∈S:the optimal strategy ˆ

Xτ,ζ,φ ∈Xdsatisfies

ϱ∇↓

0Jτ(ˆ

Xτ,ζ,φ) = 0 for some ϱ

and ˆ

Xτ,ζ,φ,↓

0= 0}.

(2.23)

3. We define the waiting-region as

Rwait ≜S\(¯

Rbuy ∪¯

Rsell)(2.24)

where ¯

Rbuy/sell ≜Rbuy/sell ∪∂Rbuy/sell, respectively.

2 Optimal Investment with Transient Price Impact

Moreover, the formulation of our minimization problem in (2.13) as an optimal

tracking problem of the constant Merton portfolio suggests to introduce the so-

called Merton plane in the state space S.

Definition 2.3.11 (Merton plane).We define the Merton plane as

M≜{(τ, ζ, φ)∈S:φ=φ0=µ

ασ2}.(2.25)

Remark 2.3.12.Let us briefly elaborate on the Merton plane in the state space S.

Roughly speaking, it seems reasonable to expect that the Merton plane is embedded

in the waiting-region Rwait in the state space, particularly when the time horizon

is quite long. That is, optimal strategies ˆ

Xτ,ζ,φ with problem data (τ, ζ, φ)whose

time to maturity τis very large and whose initial position φis close to the Merton

position φ0are expected to remain close to the latter by simply being inactive at

the beginning. Moreover, if the initial risky asset position φis far away from the

preferred Merton portfolio, we expect that the optimal strategy will immediately

start trading towards φ0as long as the initial spread ζis not too high; otherwise we

estimate that the optimizer mitigates trading costs by benefiting from the resilience

effect and does not start trading directly but waits until the initial spread ζhas

sufficiently decreased; cf. the dynamics in (2.3). Therefore, it is natural to expect

that the selling-region Rsell will be located above the Merton plane Min the sense

that (τ, ζ, φ)∈Rsell implies φ≥µ/(ασ2)whereas the buying-region Rbuy will be

below the Merton plane M, i.e., it holds that φ≤µ/(ασ2)for all (τ, ζ, φ)∈Rbuy.

Indeed, our analysis will show that the latter holds true for any τ≥0and any

ζ≥0. However, due to the fact that the optimal strategy ˆ

Xτ,ζ,φ also has to

take into account the ultimate liquidation of any non-zero position φ= 0 in the

risky asset when approaching terminal time, it turns out that the picture changes

significantly in case where the planning horizon τshortens. In fact, there are

triplets (τ, ζ, φ)∈Rsell satisfying φ < µ/(ασ2)if τand ζare small enough. In other

words, the corresponding optimizer ˆ

Xτ,ζ,φ prefers to immediately start unwinding

his initial asset position φand simply ignores the tracking of the original target

φ0=µ/(ασ2).

For the rest of this section, let us collect some properties of the buying- and

selling-region as well as their boundaries. First of all, note that Lemma 2.3.8

implies ¯

Rbuy ∩¯

Rsell =∅, that is, the boundaries of the buying- and selling-region

do not touch in the state space S. The next Lemma 2.3.13 justifies their definitions

in (2.21) and (2.23). It states that optimal strategies with problem data (τ, ζ, φ)∈

Swhich belong to the buying-region Rbuy (selling-region Rsell) and thus start by

definition in (2.20) (in (2.22)) with an initial impulse block buy (sell) order, will

actually “jump” to the boundary of the buying region ∂Rbuy (selling region ∂Rsell).

Note that this initial jump widens the initial spread ζto ζ+ηx, cf. the dynamics

of the spread in (2.3), and increases (decreases) the initial position from φto φ+x

(φ−x).

2.3 Case study: Illiquid Bachelier model with exponential utility

Lemma 2.3.13 (Initial impulse extension).

1. Let (τ, ζ, φ)∈∂Rbuy be some arbitrary problem data. Then for any data

(τ, ζ −ηx, φ −x)∈Swith 0< x ≤ζ/η the corresponding optimal strategy

is given by

Xτ,ζ−ηx,φ−x= (x+ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)∈Xd.(2.26)

In particular, (τ, ζ −ηx, φ −x)∈Rbuy.

2. Analogously, let (τ, ζ, φ)∈∂Rsell be some arbitrary problem data. Then for

any data (τ, ζ −ηx, φ +x)∈Swith 0< x ≤ζ/η the corresponding optimal

strategy is given by

Xτ,ζ−ηx,φ+x= ( ˆ

Xτ,ζ,φ,↑, x +ˆ

Xτ,ζ,φ,↓)∈Xd.(2.27)

In particular, (τ, ζ −ηx, φ +x)∈Rsell.

Put differently, in view of the dynamic programming principle from Remark 2.3.9,

any optimal strategy ˆ

Xτ,ζ,φ with problem data (τ, ζ, φ)∈∂Rbuy or ∂Rsell can be

“extended” instantaneously with a block trade at time 0 to obtain an optimal policy

for any valid jump size x∈(0, ζ/η]. The next lemma characterizes the behaviour

of optimal strategies ˆ

Xτ,ζ,φ with data (τ, ζ, φ)belonging to the boundary of the

buying region ∂Rbuy (selling region ∂Rsell) which continue buying (selling) steadily

until some future point in time.

Lemma 2.3.14. Let (τ, ζ, φ)∈Swith τ > 0be some arbitrary data with corre-

sponding optimal strategy ˆ

Xτ,ζ,φ ∈Xd. Assume that

[0, v]⊂ {dˆ

Xτ,ζ,φ,↑>0} ⊂ {ϱ∇↑Jτ(ˆ

Xτ,ζ,φ,↑) = 0}for some ϱ

and 0< v ≤τ. Then, the process ˆ

Xτ,ζ,φ,↑is absolutely continuous on (0, v). In

particular, it holds that

(τ−t, ζ ˆ

Xτ,ζ,φ

t, φ ˆ

Xτ,ζ,φ

t)∈∂Rbuy (0 < t < v).(2.28)

The same assertion holds true when interchanging ↑by ↓.

In other words, Lemma 2.3.14 shows that for any data (τ, ζ, φ)∈∂Rbuy (or

∂Rsell) such that the associated optimal strategy ˆ

Xτ,ζ,φ immediately continues

buying (selling) the risky asset until some future point in time v∈(0, τ]will do

this continuously on [0, v). As a consequence, the trajectory of the corresponding

optimally controlled triplet in (2.28) will stay on the boundary during this buying

period (selling period). The next lemma provides that this “sliding” on the buying

boundary (selling boundary) induces a particular dynamic for the corresponding

optimally controlled share holdings φˆ

Xτ,ζ,φ .

2 Optimal Investment with Transient Price Impact

Lemma 2.3.15 (Boundary dynamics).Let τ > 0and let ˆ

X= ( ˆ

X↑,ˆ

X↓)∈Xdbe

an optimal strategy minimizing Jτin (2.13)such that

[u, v]⊂ {dˆ

X↑>0} ⊂ {ϱ∇↑Jτ(ˆ

X) = 0}for some ϱ

[u, v]⊂ {dˆ

X↓>0} ⊂ {ϱ∇↓Jτ(ˆ

X) = 0}for some ϱ

and 0≤u < v ≤τ. Then, the corresponding optimally controlled share holdings

φˆ

Xsatisfy the second order ODE

¨φˆ

t=β2(φˆ

t−µ

ασ2)for t∈(u, v)(2.29)

with

β≜κ

√1 + κη

ασ2

.(2.30)

Obviously, Lemma 2.3.15 will be very helpful in identifying candidates for those

optimal strategies ˆ

Xwhich buy (sell) continuously during some time period and

therefore the controlled state process evolves in the boundary of the buying (selling)

region in the sense of (2.28) in Lemma 2.3.14. So far as the above ODE will not

in general hold on the entire time interval [0, τ], but merely on a subinterval to be

determined with proper boundary conditions.

To sum up, in order to describe buying-, waiting- and selling-region in the state

space S, that is, the sets Rbuy,Rwait and Rsell from Definition 2.3.10 above, we es-

pecially have to identify the free boundaries ∂Rbuy and ∂Rsell. It turns out that the

characterization of ∂Rsell is rather simple and fully described by a smooth surface

in S. This will be done in Section 2.3.3. The characterization of the free bound-

ary of the buying region ∂Rbuy is much more involved, though. It particularly

requires the identification of the proper boundary and the corresponding boundary

conditions in the state space Sto the ODE (2.29) in Lemma 2.3.15. This anal-

ysis is carried out in Section 2.3.4. A synopsis of the results will be presented in

Section 2.3.5.

Remark 2.3.16.Observe that by the definition in (2.19) the problem data or triplets

(0, ζ, φ)with τ= 0 also belong to the state space S. Hence, we have to find a con-

vention for specifying the associated optimal strategies ˆ

X0,ζ,φ satisfying φˆ

X0,ζ,φ

0−=φ

and ζˆ

X0,ζ,φ

0−=ζ. In view of the subgradients in (2.15) and (2.16) we have

ϱ∇↑,↓

0J0(ˆ

X0,ζ,φ) = 1

2(η|φ|+ζ)±η

2φ±1

2signϱ(φ)ζ.

Thus, in case φ > 0it holds that ∇↑

0J0(ˆ

X0,ζ,φ)>0and ∇↓

0J0(ˆ

X0,ζ,φ)=0. There-

fore, we stipulate that the associated optimal strategy ˆ

X0,ζ,φ is given by ˆ

X0,ζ,φ,↑

0≜0

and ˆ

X0,ζ,φ,↓

0≜φ > 0, i.e., it unwinds with a single block sell order the position φ.

Analogously, in case φ < 0we have ∇↑

0J0(ˆ

X0,ζ,φ) = 0 and ∇↓

0J0(ˆ

X0,ζ,φ)>0and

2.3 Case study: Illiquid Bachelier model with exponential utility

thus we set ˆ

X0,ζ,φ,↓

0≜0as well as ˆ

X0,ζ,φ,↑

0=−φ > 0, i.e., the optimal strategy

evens its position by executing a single block buy order. In case φ= 0, we have

ϱ∇↑,↓

0J0(ˆ

X0,ζ,0) = 1

2ζ±1

2ϱζ ≥0

for all ϱ∈[−1,1]. We make the convention that the associated optimal strategy is

simply defined as ˆ

X0,ζ,0

0≜(0,0).

2.3.3 Characterization of the free boundary of the selling-region

For ease of presentation, let us introduce the constants

λ≜√ασ, γ±≜λ±√κη +λ2(2.31)

as well as for all τ≥0the parameter C(τ), D(τ)given by

C(τ)≜e−βτ γ−+eβτ γ+

e−βτ γ2

−+eβτ γ2

, D(τ)≜1−2κη

e−βτ γ2

−+eβτ γ2

.(2.32)

The next proposition characterizes all triplets (τ, ζ, φ)in the state space Swhose

corresponding optimal strategies ˆ

Xτ,ζ,φ sell continuously during the entire time

period [0, τ].

Proposition 2.3.17. Let (τ, ζ, φ)∈S,τ > 0, satisfy

φ=ϕ1(τ, ζ)≜µ

λ2D(τ) + κ

λζC(τ).(2.33)

Then the optimal strategy ˆ

Xτ,ζ,φ ∈Xdwith problem data (τ, ζ, φ)is given by

Xτ,ζ,φ,↑

t= 0 and ˆ

Xτ,ζ,φ,↓

t=ℓt(τ, ζ, φ)for all 0≤t≤τwith

ℓt(τ, ζ, φ)≜c+(τ, ζ, φ)eβt +c−(τ, ζ, φ)e−βt +φ−µ

λ2(2.34)

where

c±(τ, ζ, φ)≜κ(e∓βτ γ∓(ηµ −λ2(ζ+ηφ)) + ηµγ±)

λ2√κη +λ2(eβτ γ2

+−e−βτ γ2

−)(2.35)

and βas defined in (2.30). In particular, it holds that (τ, ζ, φ)∈∂Rsell, the optimal

share holdings φˆ

Xτ,ζ,φ satisfy the second order ODE in (2.29)on (0, τ)and we have

φˆ

Xτ,ζ,φ

t=ϕ1(τ−t, ζ ˆ

Xτ,ζ,φ

t)>0 (0 ≤t≤τ).(2.36)

Observe that the optimal cumulative sales in (2.34) are absolutely continuous on

[0, τ], that is, no impulse sell orders occur. By virtue of Lemma 2.3.13, we obtain

the following corollary.

2 Optimal Investment with Transient Price Impact

Corollary 2.3.18. Let (τ, ζ, φ)∈Ssatisfy

φ > ϕ1(τ, ζ).(2.37)

Set

x≜φ−ϕ1(τ, ζ)

1 + κ

ληC(τ)>0.(2.38)

Then the optimal strategy ˆ

Xτ,ζ,φ with problem data (τ, ζ, φ)is given by

Xτ,ζ,φ,↑

t= 0,ˆ

Xτ,ζ,φ,↓

t=x+ˆ

Xτ,ζ+ηx,φ−x,↓

ton [0, τ](2.39)

where ˆ

Xτ,ζ+ηx,φ−x

·= (0, ℓ·(τ, ζ +ηx, φ −x)) denotes the optimal strategy from

Proposition 2.3.17 with problem data (τ, ζ +ηx, φ −x). In particular, we have

(τ, ζ, φ)∈Rsell.

Note that the initial block sell order xin (2.38) of the optimal strategy ˆ

Xτ,ζ,φ

from Corollary 2.3.18 is chosen so as to satisfy the identity

φ−x=ϕ1(τ, ζ +ηx).(2.40)

In other words, in line with Lemma 2.3.13, the optimal strategy ˆ

Xτ,ζ,φ from Corol-

lary 2.3.18 with data (τ, ζ, φ)∈Rsell “jumps” with an initial impulse trade of

size xto the boundary triplet (τ, ζ +ηx, φ −x)∈∂Rsell satisfying the identity in

(2.33) and then follows the associated optimal strategy ˆ

Xτ,ζ+ηx,φ−xfrom Proposi-

tion 2.3.17. Put differently, appealing to Lemma 2.3.14 and Lemma 2.3.15 above,

the free boundary ∂Rsell described in (2.33) is completely traced out by the evolu-

tion in Sof the optimally controlled state processes, that is, by the dynamics of

the optimal share holdings in (2.36) satisfying the second order ODE in (2.29) on

[0, τ]for any τ > 0as well as their corresponding spread dynamics.

Remark 2.3.19.

1. Concerning the optimal strategy ˆ

Xτ,ζ,φ from Proposition 2.3.17, note that

property (2.36) implies φˆ

Xτ,ζ,φ

t>0for all t∈[0, τ]. In particular, for the

remaining shares φˆ

Xτ,ζ,φ

τin the risky asset at final time τit holds that

φˆ

Xτ,ζ,φ

τ=ϕ1(0, ζ ˆ

Xτ,ζ,φ

τ) = 2µ+κζ ˆ

Xτ,ζ,φ

2λ2+κη >0(2.41)

due to Lemma 2.4.2. Hence, in line with our convention in Remark 2.3.16,

we will assume that the optimal strategy ˆ

Xτ,ζ,φ from Proposition 2.3.17 liq-

uidates at final time τthe remaining shares φˆ

Xτ,ζ,φ

τwith a single block sell

order.

2. Observe that the condition in (2.33) or (2.37) on the triplet (τ, ζ, φ)∈Sdoes

not necessarily imply φ≥φ0=µ/λ2. Moreover, it is interesting to note that

independently of the planning horizon τthe optimal strategies ˆ

Xτ,ζ,φ from

Proposition 2.3.17 trade and sell steadily all along [0, τ]even when the initial

holdings φexceeding the preferred Merton position φ0have been reduced to

the latter in the meantime.

2.3 Case study: Illiquid Bachelier model with exponential utility

3. In case ζ=µ= 0, the optimal strategy from Corollary 2.3.18 together

with the convention in 1.) above coincides for any φ > 0with the optimal

liquidation strategy computed in Obizhaeva and Wang (2013), Proposition 4.

2.3.4 Characterization of the free boundary of the buying-region

Our goal in this section is to identify the free boundary of the buying-region ∂Rbuy

in the state space Sas defined in (2.21). It turns out that this task is more involved

compared to the free boundary of the selling-region ∂Rsell in Section 2.3.3 above.

Specifically, motivated by Lemma 2.3.14 and Lemma 2.3.15 we have to identify

the proper domain as well as the corresponding proper boundary conditions for

the ODE in (2.29). As it turns out, we will detect an associated free boundary

curve in the free boundary ∂Rbuy in the three-dimensional state space Svia a

parametrization in three properly chosen parameters, namely τ,θand ϱwhere the

role of the new variable θwill be explained below. Similar to the free boundary

of the selling-region described in Proposition 2.3.17, a major part of ∂Rbuy will

then be traced out by the evolution in Sof the controlled state processes, that

is, the dynamics of the share holdings satisfying the second order ODE in (2.29)

with the identified boundary conditions on the free boundary curve as well as

their corresponding spread dynamics. Beyond that and contrary to ∂Rsell a minor

part of ∂Rbuy will additionally be characterized by stand-alone initial impulse buy

trades.

Let us briefly describe our methodology. Loosely speaking, appealing to the

dynamic programming principle from Remark 2.3.9, we proceed via a backward

induction in time in the state space S. We start with identifying all triplets

(τ, ζ, φ)∈∂Rsell described in Proposition 2.3.17 via equation (2.33) which can be

“reached” by an optimally controlled state process evolving in Ssuch that the asso-

ciated optimal strategy originally “departed” at some prior point in time from some

problem data belonging to the boundary of the buying-region ∂Rbuy. Recall that

the boundaries of the buying- and selling-region do not touch in the state space due

to Lemma 2.3.8. Therefore, the considered trajectory necessarily passes through

the waiting region Rwait in Sdefined in (2.24). In doing so, the position in the

risky asset remains unchanged with φshares and the spread evolves according to

the uncontrolled dynamics dζt=−κζt−dt in (2.2). In other words, we have to iden-

tify those problem data (τ, ζ, φ)∈∂Rsell and those θ > 0such that (τ+θ, ζeκθ, φ)

belongs to ∂Rbuy and allows for an associated optimal strategy ˆ

Xτ+θ,ζeκθ,φ which

does not trade until time θand then follows the optimal strategy corresponding to

the triplet (τ, ζ, φ)in ∂Rsell. In fact, because of Lemma 2.4.4 2.) below, it suffices

to start with triplets (τ, ζ, φ)∈∂Rsell satisfying equation (2.33) whose position

in the risky asset φis strictly below the Merton portfolio φ0=µ/λ2=µ/(ασ2).

After having examined all these points in the above described manner, we continue

investigating by the same methodology all problem data (0, ζ, φ)∈Swith τ= 0,

2 Optimal Investment with Transient Price Impact

ζ≥0and φ∈R.

2.3.4.1 Parametrization of the free boundary curve in τ

Let (τ, ζ, φ)∈∂Rsell,τ > 0, be some arbitrary problem data such that

0< φ =ϕ1(τ, ζ)< φ0=µ

λ2=µ

ασ2(2.42)

with ϕ1as defined in (2.33). Hence, the strictly positive number of shares φin

the risky asset is strictly below the Merton position φ0. Let ˆ

Xτ,ζ,φ denote the

corresponding optimal sell-only strategy from Proposition 2.3.17 given by (2.34).

Furthermore, we explicitly want to allow for triplets (0, ζ, φ)∈Swith τ= 0

satisfying (2.42). In this case, we follow our convention from Remark 2.3.16 and set

X0,ζ,φ

0≜(0, φ)for the associated optimal strategy. As described at the beginning of

this section above, we make the following ansatz: For θ≥0we consider the problem

data (τ+θ, ζeκθ, φ)∈Swith associated candidate strategy Xτ+θ,ζeκθ,φ ∈Xd

which we define as

Xτ+θ,ζeκθ,φ

t≜(0,ˆ

Xτ,ζ,φ,↓

t−θ1[θ,τ+θ](t))(0 ≤t≤τ+θ)(2.43)

with φXτ+θ,ζeκθ,φ

0−≜φand ζXτ+θ,ζeκθ,φ

0−≜ζeκθ. That is, the strategy in (2.43)

for the longer time horizon τ+θis inactive until time θand then coincides with

the optimal selling strategy ˆ

Xτ,ζ,φ from Proposition 2.3.17 (or with ˆ

X0,ζ,φ

0in the

case τ= 0). In particular, it holds that φXτ+θ,ζeκθ,φ

θ−=φ=φˆ

Xτ,ζ,φ

0−as well as

ζXτ+θ,ζeκθ,φ

θ−=ζ=ζXτ,ζ,φ

0−. The idea is now to identify those triplets (τ, ζ, φ)

satisfying (2.42), including the case τ= 0, which allow for a largest θ≥0such

that the candidate strategy in (2.43) is optimal and satisfies

∇↑

0Jτ+θ(Xτ+θ,ζeκθ,φ) = 0.

In other words, since Xτ+θ,ζeκθ,φ

0= (0,0) by the definition in (2.43), the problem

data (τ+θ, ζeκθ, φ)belongs to ∂Rbuy. This is achieved by the next proposition.

Proposition 2.3.20. Let τ≥0be arbitrary.

1. Let ¯

θ > 0denote the unique positive solution to the equation

eκ¯

θ(2 −κ¯

θ) + 2 + κ¯

θ= 0.(2.44)

Set

ζ≜s1(τ)≜µ(1 −D(τ))

λκC(τ) + 1

2κeκ¯

θ(1 −e−κ¯

θ)>0,(2.45)

¯φ≜ϕ1(τ, ¯

ζ) = µ

λ2−1

2κ¯

ζeκ¯

θ1−e−κ¯

λ2>0(2.46)

2.3 Case study: Illiquid Bachelier model with exponential utility

with ϕ1as defined in (2.33). Then the problem data (τ, ¯

ζ, ¯φ)satisfies (2.42),

i.e., belongs to ∂Rsell (unless τ= 0), and the candidate strategy Xτ+¯

θ,¯

ζeκ¯

θ,¯φ

in (2.43)with problem data (τ+¯

θ, ¯

ζeκ¯

θ,¯φ)is optimal and satisfies

∇↑

0Jτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) =0 and

dt (∇↑

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ))⏐⏐⏐⏐t↓0

=0.(2.47)

In particular, (τ+¯

θ, ¯

ζeκ¯

θ,¯φ)∈∂Rbuy.

2. For all 0< ζ < ¯

ζand corresponding unique τζ≥0such that ¯φ=ϕ1(τζ, ζ),

i.e., (τζ, ζ, ¯φ)satisfies (2.42)and belongs to ∂Rsell (unless τζ= 0), denote

by θ∗∈[0,¯

θ)the unique root of the function w3(·, ζ, ¯φ, τζ)given in (2.152).

Then the candidate strategy Xτζ+θ∗,ζeκθ∗,¯φin (2.43)with problem data (τζ+

θ∗, ζeκθ∗,¯φ)is optimal and satisfies

∇↑

0Jτζ+θ∗(Xτζ+θ∗,ζeκθ∗,¯φ) =0 and

dt (∇↑

tJτζ+θ∗(Xτζ+θ∗,ζeκθ∗,¯φ))⏐⏐⏐⏐t↓0

>0.(2.48)

Again, we have (τζ+θ∗, ζeκθ∗,¯φ)∈∂Rbuy.

3. For all ζ > ¯

ζand corresponding unique τζsuch that ¯φ=ϕ1(τζ, ζ), i.e.,

(τζ, ζ, ¯φ)satisfies (2.42)and belongs to ∂Rsell (unless τζ= 0), the candidate

strategy Xτζ+θ,ζeκθ,¯φin (2.43)with problem data (τζ+θ, ζeκθ,¯φ)is optimal

for any θ > 0and satisfies

∇↑

0Jτζ+θ(Xτζ+θ,ζeκθ,¯φ)>0.(2.49)

In particular, (τζ+θ, ζeκθ,¯φ)∈Rwait for all θ > 0.

Note that the problem data (τ+¯

θ, s1(τ)eκ¯

θ, ϕ1(τ, s1(τ))) ∈∂Rbuy from Proposi-

tion 2.3.20 1.) are fully parametrized in τ≥0. Moreover, it can be easily checked

that the map τ↦→ s1(τ)is strictly decreasing in τ; cf. Lemma 2.4.5 below. In fact,

this resulting curve τ↦→ (τ+¯

θ, s1(τ)eκ¯

θ, ϕ1(τ, s1(τ))) in the three-dimensional state

space as a function in τprovides the first bit of the free boundary curve in ∂Rbuy

which is related to the unknown domain and its proper boundary conditions of

the ODE motivated in Lemma 2.3.15 in the following sense: For any τ≥0the

optimal strategy ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φfrom Proposition 2.3.20 1.) can be extended for any

ϑ > 0to obtain an optimal strategy with triplet (τ+¯

θ+ϑ, ζ′, φ′)that initially buys

the risky asset at an optimal rate until time ϑand then follows the optimal strat-

egy ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φon the remaining time interval [ϑ, τ +¯

θ+ϑ]. More precisely, due

to the dynamic programming principle from Remark 2.3.9, this optimal strategy

2 Optimal Investment with Transient Price Impact

Xτ+¯

θ+ϑ,ζ′,φ′coincides with the strategy ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φon [ϑ, τ +¯

θ+ϑ]in the sense

that

Xτ+¯

θ+ϑ,ζ′,φ′

t=ˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−+ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φ

t−ϑ(ϑ≤t≤τ+¯

θ+ϑ).(2.50)

Consequently, in view of Lemma 2.3.15, the dynamics of the corresponding opti-

mally controlled share holdings φˆ

Xτ+¯

θ+ϑ,ζ′,φ′of this extended strategy satisfy the

second order ODE in (2.29) on (0, ϑ)and they coincide at time ϑwith ¯φon the

free boundary curve. As does the corresponding spread with ¯

ζeκ¯

θ, that is, it holds

that

φˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−= ¯φand ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−=¯

ζeκ¯

θ.

The strategies in Proposition 2.3.20 2.) can be extended instantaneously at the

initial time of the period [0, τζ+θ∗]by some proper block-buy order in the sense

of Lemma 2.3.13 1.). The problem data (τζ, ζ, ¯φ)∈∂Rsell (unless τζ= 0) charac-

terized in Proposition 2.3.20 3.) do not contributed to ∂Rbuy.

Corollary 2.3.21. We define the function

c(ζ, φ)≜β2

λ2(ζ+1

κ(λ2φ−µ))(2.51)

with βas defined in (2.30). For any ϑ > 0the optimal strategy ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φfrom

Proposition 2.3.20 1.) with problem data (τ+¯

θ, ¯

ζeκ¯

θ,¯φ)∈∂Rbuy can be extended to

obtain an optimal strategy ˆ

Xτ+¯

θ+ϑ,ζ′,φ′∈Xdwith problem data (τ+¯

θ+ϑ, ζ′, φ′)∈

∂Rbuy such that

φˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−= ¯φand ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−=¯

ζeκ¯

θ.(2.52)

On the interval [0, ϑ], this strategy is given by

Xτ+¯

θ+ϑ,ζ′,φ′,↓

t≜0,

Xτ+¯

θ+ϑ,ζ′,φ′,↑

t≜1

2(¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)(e−β(ϑ−t)−e−βϑ)

2(¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)(eβ(ϑ−t)−eβϑ).

(2.53)

The pair (ζ′, φ′)is given by

ζ′≜¯

ζeκ(¯

θ+ϑ)−η∫ϑ

eκudˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑

u>¯

ζeκ¯

θ,

φ′≜(¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)e−βϑ

+(¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)eβϑ

2+µ

λ2∈R.

(2.54)

2.3 Case study: Illiquid Bachelier model with exponential utility

In particular, the optimal share holdings φˆ

Xτ+¯

θ+ϑ,ζ′,φ′satisfy the second order ODE

in (2.29)on (0, ϑ)and we have

(τ+¯

θ+ϑ−t, ζ ˆ

Xτ+¯

θ+ϑ,ζ′,φ′

t, φ ˆ

Xτ+¯

θ+ϑ,ζ′,φ′

t)∈∂Rbuy (0 ≤t≤ϑ).(2.55)

Remark 2.3.22.

1. The optimal investment strategies obtained in Corollary 2.3.21 are char-

acterized by a buying, waiting and selling period on [0, ϑ],[ϑ, ¯

θ+ϑ]and

[¯

θ+ϑ, τ +¯

θ+ϑ], respectively, as long as τ > 0. Note that the associated

initial position φ′in (2.54) in the risky asset takes a value in Rand therefore

may be negative. Interestingly, the length of the waiting period is always

given by the constant value ¯

θsatisfying the identity in (2.44). In contrast, re-

call that the optimal investment strategies described in Proposition 2.3.20 2.)

only allow for an initial block buy order at the beginning of the waiting period

[0, θ∗](due to Lemma 2.3.13 1.)) and then sell the risky asset on [θ∗, τζ+θ∗]

at some optimal rate unless τζ= 0. In both cases, note that the accrued

position in the risky asset ¯φgiven in (2.46) after the buying period [0, ϑ]or,

in the latter case, after the initial block buy order is always strictly below the

Merton-portfolio φ0=µ/λ2.

2. Obviously, Lemma 2.3.13 1.) is applicable to any problem data in (2.55).

Hence, together with the convention in Remark 2.3.19 1.), the optimal invest-

ment strategies characterized in Corollary 2.3.21 on the interval [0, τ +¯

θ+ϑ]

only allow for block buy and sell trades at the beginning and the end of the

considered investment period but not in between. This also holds true for the

optimal investment strategies described in Proposition 2.3.20 2.) on [0, τζ+θ∗]

and is similar to the optimal liquidation strategies computed in Obizhaeva

and Wang (2013). Concerning the optimal investment strategies described in

Proposition 2.3.20 3.) on [0, τζ+θ]for any θ > 0, an initial block buy order

is not optimal, though, because the initial spread is too high.

3. In the proof of Corollary 2.3.21 it is shown that the controlled spread ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

is actually strictly decreasing on the interval [0, ϑ]which in particular implies

that ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

0−=ζ′>¯

ζeκ¯

θas claimed in (2.54). Put differently, the optimal

strategy described in Corollary 2.3.21 exploits the resilience effect during the

buying period [0, ϑ]in the sense that the spread recovers at a faster rate given

by κthan the rate at which the absolutely continuous optimizer purchases

the risky asset on [0, ϑ].

4. Let us also mention that Lemma 2.4.4 1.) is applicable to any problem data

in (2.55) because ¯φin (2.46) is below the Merton position.

2 Optimal Investment with Transient Price Impact

2.3.4.2 Parametrization of the free boundary curve in θ

In the case τ= 0, we obtain in Proposition 2.3.20 1.) above the problem data

(¯

θ, s1(0)eκ¯

θ, ϕ1(0, s1(0))) ∈∂Rbuy via the triplet (0, s1(0), ϕ1(0, s1(0))) ∈Swhich

satisfies the identity

ϕ1(0, s1(0)) = 2µ+κs1(0)

2λ2+κη

due to the definition in (2.45). This motivates to pursue our goal of describing the

boundary of the buying region ∂Rbuy by investigating with the same methodology

as above all triplets (0, ζ, φ)∈Swhich satisfy

0≤φ≤2µ+κζ

2λ2+κη.(2.56)

Specifically, let (0, ζ, φ)∈Ssatisfy the bounds in (2.56). Again, as before, fol-

lowing our convention from Remark 2.3.16, we denote by ˆ

X0,ζ,φ the corresponding

optimal strategy which is simply defined as

X0,ζ,φ,↑

0≜0,ˆ

X0,ζ,φ,↓

0≜φ≥0.

Next, analogously to (2.43) above, we consider for any θ≥0the problem data

(θ, ζeκθ, φ)with associated candidate strategy Xθ,ζeκθ,φ ∈Xdwhich is now defined

Xθ,ζeκθ,φ

t≜(0, φ1{θ}(t))(0 ≤t≤θ).(2.57)

In other words, the candidate strategy in (2.57) is inactive until time θand then

coincides with the optimal strategy ˆ

X0,ζ,φ, i.e., liquidates with a single block sell

order the shares φ≥0. Similar to Proposition 2.3.20, the next proposition provides

a classification of all problem data (0, ζ, φ)∈Ssatisfying (2.56) which allow for a

largest θ≥0such that the candidate strategy in (2.57) is optimal and satisfies

1∇↑

0Jθ(Xθ,ζeκθ,φ) = 0,(2.58)

i.e., (θ, ζeκθ, φ)∈∂Rbuy. Recall that the left-hand superscript in the operator

symbol of the buy-subgradient in (2.58) represents the value ϱ= signϱ(0), see

Remark 2.3.5. Since we explicitly allow for φ= 0 in (2.56), we have to be precise

about this value. In this subsection we set ϱ= 1.

Proposition 2.3.23. Let θ∈[θ,¯

θ]be arbitrary, where ¯

θ > 0satisfies the equation

in (2.44)and θ∈(0,¯

θ)denotes the unique solution to the equation

eκθ(κθ −1) = 1.(2.59)

1. Set

ζ≜s2(θ)≜µηe−κθ κθe−κθ +1+e−κθ

κλ2θ+1

2κη(1 + e−κθ)2−λ2(1 + e−κθ)>0,(2.60)

¯φ≜ϕ2(θ)≜µκθ −µ(1 + e−κθ)

κλ2θ+1

2κη(1 + e−κθ)2−λ2(1 + e−κθ)≥0.(2.61)

2.3 Case study: Illiquid Bachelier model with exponential utility

Then, the problem data (0,¯

ζ, ¯φ)∈Ssatisfies (2.56)and the candidate strat-

egy Xθ,¯

ζeκθ,¯φin (2.57)with problem data (θ, ¯

ζeκθ,¯φ)is optimal and satisfies

1∇↑

0Jθ(Xθ,¯

ζeκθ,¯φ) =0 and

dt (1∇↑

tJθ(Xθ,¯

ζeκθ,¯φ))⏐⏐⏐⏐t↓0

=0.(2.62)

In particular, (θ, ¯

ζeκθ,¯φ)∈∂Rbuy.

2. For all ζ∈(max{0,( ¯φ(2λ2+κη)−2µ)/κ},¯

ζ)the triplet (0, ζ, ¯φ)satisfies (2.56).

Denote by θ∗∈[0, θ)the unique root of the function w3(·, ζ, ¯φ, 0) given in

(2.174). Then the candidate strategy Xθ∗,ζeκθ∗,¯φin (2.57)with problem data

(θ∗, ζeκθ∗,¯φ)is optimal and satisfies

1∇↑

0Jθ∗(Xθ∗,ζeκθ∗,¯φ) =0 and

dt (1∇↑

tJθ∗(Xθ∗,ζeκθ∗,¯φ))⏐⏐⏐⏐t↓0

>0.(2.63)

Again, we have (θ∗, ζeκθ∗,¯φ)∈∂Rbuy.

3. For all ζ > ¯

ζthe triplet (0, ζ, ¯φ)satisfies (2.56)and the candidate strategy

Xθ,ζeκθ,¯φin (2.57)with problem data (θ, ζeκθ,¯φ)is optimal for any θ > 0and

satisfies

1∇↑

0Jθ(Xθ,ζeκθ,¯φ)>0.(2.64)

In particular, (θ, ζeκθ,¯φ)∈Rwait for all θ > 0.

Obviously, the problem data (θ, ¯

ζeκθ,¯φ) = (θ, s2(θ)eκθ, ϕ2(θ)) ∈∂Rbuy from

Proposition 2.3.23 1.) are fully parametrized in θ∈[θ, ¯

θ]. In fact, the map

θ↦→ s2(θ)in (2.60) is strictly decreasing and the map θ↦→ ϕ2(θ)in (2.61) is

strictly increasing on [θ, ¯

θ]; cf. Lemma 2.4.6 1.) below. Moreover, for θ=¯

θ,

the obtained triplet (¯

θ, s2(¯

θ)eκ¯

θ, ϕ2(¯

θ)) ∈∂Rbuy actually coincides with the triplet

(¯

θ, s1(0)eκ¯

θ, ϕ1(0, s1(0))) ∈∂Rbuy from Proposition 2.3.20 1.) in the case τ= 0;

again cf. Lemma 2.4.6 2.). Put differently, we have detected another bit of the un-

known free boundary curve in ∂Rbuy via a parametrization in θ. In fact, analogously

to Corollary 2.3.21 above, the optimal strategy ˆ

Xθ,¯

ζeκθ,¯φfrom Proposition 2.3.20 1.)

can be extended for any ϑ > 0to obtain an optimal strategy on the longer time

horizon θ+ϑwhich continuously buys the risky asset until time ϑand then fol-

lows ˆ

Xθ,¯

ζeκθ,¯φon [ϑ, θ +ϑ]. That is, the dynamics of the corresponding optimally

controlled share holdings of this extended strategy solve the second order ODE in

(2.29) on (0, ϑ)and then coincide at time ϑwith the value ¯φon the free boundary

curve as does the corresponding spread with the value ¯

ζeκθ. As above, the strate-

gies ˆ

Xθ∗,ζeκθ∗,¯φdescribed in Proposition 2.3.23 2.) can be extended instantaneously

with an initial block buy order in the sense of Lemma 2.3.13. The problem data

(0, ζ, ¯φ)∈Scharacterized in Proposition 2.3.23 3.) do not contribute to ∂Rbuy.

2 Optimal Investment with Transient Price Impact

Corollary 2.3.24. For any ϑ > 0, the optimal strategy ˆ

Xθ,¯

ζeκθ,¯φfrom Proposition

2.3.23 1.) with problem data (θ, ¯

ζeκθ,¯φ)∈∂Rbuy can be extended to obtain an

optimal strategy ˆ

Xθ+ϑ,ζ′,φ′∈Xdwith problem data (θ+ϑ, ζ′, φ′)∈∂Rbuy such

that

φˆ

Xθ+ϑ,ζ′,φ′

ϑ−= ¯φand ζˆ

Xθ+ϑ,ζ′,φ′

ϑ−=¯

ζeκθ.(2.65)

On the interval [0, ϑ], this strategy as well as the pair (ζ′, φ′)are given as in Corol-

lary 2.3.21 in (2.53)and (2.54), respectively, with τ= 0,cevaluated at (¯

ζeκθ,¯φ)

and ζ′>¯

ζeκθ.

Compared to Corollary 2.3.21, the optimal investment strategies obtained in

Corollary 2.3.24 are characterized by a buying and waiting period on [0, ϑ]and

[ϑ, θ +ϑ], respectively, with a final block sell order of the accrued position ¯φ=

ϕ2(θ)≥0in (2.61) at time θ+ϑwhich is again always below the Merton-portfolio

φ0. Moreover, let us mention that the comments in Remark 2.3.22 2.), 3.) and 4.)

above apply accordingly to the optimal strategies obtained in Corollary 2.3.24.

2.3.4.3 Parametrization of the free boundary curve in ϱ

In case θ=θ>0in Proposition 2.3.23 1.) above, we obtain the problem data

(θ, s2(θ)eκθ, ϕ2(θ)) = (θ, 2µ/κ, 0) ∈∂Rbuy by virtue of Lemma 2.4.6 3.) whose

associated optimal strategy in (2.57) ends up in the triplet (0, s2(θ),0) ∈S. This

motivates to continue the above analysis by examining via the same procedure as

before all problem data (0, ζ, 0) ∈Swhere the position in the risky φis equal to

zero. In line with our convention from Remark 2.3.16, the corresponding optimal

strategy ˆ

X0,ζ,0is simply given by ˆ

X0,ζ,0,↑

0=ˆ

X0,ζ,0,↓

0= 0. Analogously to (2.57) and

(2.43) above, we let θ≥0be arbitrary and consider the problem data (θ, ζeκθ,0) ∈

Swith candidate strategy

Xθ,ζeκθ,0

t≜(0,0) (0 ≤t≤θ)(2.66)

having a constant position of zero in the risky asset during the entire time period

[0, θ]. In the same spirit as Propositions 2.3.20 and 2.3.23, the following proposition

provides a classification of all problem data (0, ζ, 0) ∈Swhich allow for a largest

θ≥0such that the candidate strategy in (2.66) is optimal and satisfies

ϱ∇↑

0Jθ(Xθ,ζeκθ,0) = 0 (2.67)

for some ϱ∈[−1,1], that is, (θ, ζeκθ,0) ∈∂Rbuy. Since φXθ,ζeκθ,0

θ= 0 at final

time θ, the buy-subgradient in (2.67) depends on ϱ= signϱ(0) in this case, recall

Remark 2.3.5. As a consequence, we obtain a family of optimal strategies of type

(2.66) which is parametrized by ϱ∈[−1,1].

Proposition 2.3.25. Let ϱ∈[−1,1] be arbitrary and let θϱ∈[0, θ]denote the

unique solution to the equation

eκθϱ(κθϱ−1) = ϱ. (2.68)

2.3 Case study: Illiquid Bachelier model with exponential utility

1. Set

ζϱ≜s3(ϱ)≜2µ

κe−κθϱ∈[2µ

κe−κθ,2µ

κ].(2.69)

Then, for the problem data (0,¯

ζϱ,0) ∈Sthe candidate strategy Xθϱ,eκθϱ¯

ζϱ,0in

(2.66)with problem data (θϱ,¯

ζϱeκθϱ,0) = (θϱ,2µ/κ, 0) is optimal and satisfies

ϱ∇↑

0Jθϱ(Xθϱ,2µ/κ,0) =0 and

dt (ϱ∇↑

tJθϱ(Xθϱ,2µ/κ,0))⏐⏐⏐⏐t↓0

=0.(2.70)

In particular, (θϱ,2µ/κ, 0) ∈∂Rbuy.

2. For all 0< ζ < ¯

ζ1=e−κθ2µ/κ =s2(θ)we are in the case of Proposition

2.3.23 2.) with θ=θand triplet (0, ζ, ϕ2(θ)) = (0, ζ, 0).

3. For all ζ > ¯

ζ−1= 2µ/κ with triplet (0, ζ, 0) ∈S, the candidate strategy

Xθ,ζeκθ,0in (2.66)with problem data (θ, ζeκθ,0) is optimal for any θ > 0and

satisfies

˜ϱ∇↑

0Jθ(Xθ,ζeκθ,0)>0for all ˜ϱ∈[−1,1].(2.71)

In particular, (θ, ζeκθ,0) ∈Rwait for all θ > 0.

In fact, the problem data (θϱ,¯

ζϱeκθϱ,0) = (θϱ,2µ/κ, 0) ∈∂Rbuy in Proposi-

tion 2.3.25 1.) are fully parametrized in ϱ∈[−1,1] by virtue of Lemma 2.4.7

below which states that θϱspecified in (2.68) is unique and strictly increasing

in ϱ. In addition, it holds that for ϱ= 1 the triplet (θ1,2µ/κ, 0) ∈∂Rbuy co-

incides with the triplet (θ, s2(θ)eκθ, ϕ2(θ)) ∈∂Rbuy from Proposition 2.3.23 1.)

with θ=θ; cf. Lemma 2.4.6 3.) and Lemma 2.4.7. Since for ϱ=−1we obtain

(θ−1,2µ/κ, 0) = (0,2µ/κ, 0) ∈∂Rbuy, it turns out that we have detected the fi-

nal bit of the unknown free boundary curve in ∂Rbuy via a parametrization in ϱ.

Specifically, analogously to Corollary 2.3.21 and 2.3.24 above, the optimal strategy

Xθϱ,2µ/κ,0from Proposition 2.3.25 1.) can be extended for any ϑ > 0to obtain

an optimal strategy on the longer time horizon θϱ+ϑwhich continuously buys

the risky asset until time ϑand then follows ˆ

Xθϱ,2µ/κ,0on [ϑ, θϱ+ϑ]. Again, the

dynamics of the corresponding optimally controlled share holdings of this extended

strategy solve the second order ODE in (2.29) on (0, ϑ)and then coincide at time ϑ

with 0on the free boundary curve as does the corresponding spread with 2µ/κ.

As before, this is particularly due to the second property in (2.70) and does not

apply to the optimal strategies described in Proposition 2.3.25 2.), i.e., in Proposi-

tion 2.3.23 2.), unlike Lemma 2.3.13. The problem data (0, ζ, 0) ∈Scharacterized

in Proposition 2.3.25 3.) do not contribute to ∂Rbuy.

Corollary 2.3.26. For any ϑ > 0the optimal strategy ˆ

Xθϱ,2µ/κ,0from Proposition

2.3.25 1.) with problem data (θϱ,¯

ζϱeκθϱ,0) = (θϱ,2µ/κ, 0) ∈∂Rbuy can be extended

2 Optimal Investment with Transient Price Impact

to obtain an optimal strategy ˆ

Xθϱ+ϑ,ζ′,φ′∈Xdwith problem data (θϱ+ϑ, ζ′, φ′)∈

∂Rbuy such that

φˆ

Xθϱ+ϑ,ζ′,φ′

ϑ−= 0 and ζˆ

Xθϱ+ϑ,ζ′,φ′

ϑ−=2µ

κ.(2.72)

On the interval [0, ϑ], this strategy as well as the pair (ζ′, φ′)are given as in Corol-

lary 2.3.21 in (2.53)and (2.54), respectively, with τ= 0,cevaluated at (2µ/κ, 0)

yielding c(2µ/κ, 0) = β2µ/(λ2κ)and ζ′>2µ/κ.

Similar to Corollary 2.3.24, the optimal investment strategies obtained in Corol-

lary 2.3.26 are characterized by a buying and waiting period on the intervals [0, ϑ]

and [ϑ, θϱ+ϑ], respectively, unless ϱ=−1which implies θ−1= 0. In contrast,

the overall purchased position in the risky asset holdings during the buying period

[0, ϑ]is always zero, that is, there is no terminal block sell order at final time.

In particular, observe that for ϱ=−1the optimal strategy from Corollary 2.3.26

continuously evens its initial negative position φ′<0exactly until time ϑ.

Interestingly, the dependence of the buy-subgradients on ϱin (2.67) induces a

“plateau” on the free boundary of the buying-region ∂Rbuy along the free boundary

curve (θϱ,2µ/κ, 0) ∈∂Rbuy obtained in Proposition 2.3.25 1.) which is parametrized

in ϱ. More precisely, let ϱ∈(−1,1]. Then, along the trajectory of the optimally

controlled triplet

(θϱ−t, 2µ

κe−κθϱt,0)0≤t≤θϱ⊂S(2.73)

of the optimal strategy ˆ

Xθϱ,2µ/κ,0in (2.66) from Proposition 2.3.25 1.) with problem

data (θϱ,2µ/κ, 0) ∈∂Rbuy, it holds by construction that

ϱ∇↑

tJθϱ(ˆ

Xθϱ,2µ/κ,0)>0 (0 < t ≤θϱ);

cf. also the proof of Proposition 2.3.25. However, for every t∈(0, θϱ]there exists

a value ¯ϱ∈[−1, ϱ)such that

¯ϱ∇↑

tJθϱ(ˆ

Xθϱ,2µ/κ,0) = 0

holds true. Put differently, by the definition of ∂Rbuy in (2.21) this implies that

(θϱ−t, 2µ

κe−κθϱt,0)∈∂Rbuy (0 ≤t≤θϱ)

although the optimal strategy ˆ

Xθϱ,2µ/κ,0is inactive on [0, θϱ], i.e., Lemma 2.3.14

does not apply. Actually, the same holds true for all optimal strategies ˆ

Xθ∗,ζeκθ∗,0

from Proposition 2.3.23 2.) with θ=θ,θ∗< θ and ¯ϱ∈[−1,1).

2.3 Case study: Illiquid Bachelier model with exponential utility

Proposition 2.3.27 (Plateau).

1. Let ϱ∈(−1,1] be arbitrary but fixed and let θϱas well as ¯

ζϱ∈[e−κθ12µ/κ, 2µ/κ)

denote the corresponding quantities from Proposition 2.3.25 1.). Then the

following holds true: For all ¯ϱ∈[−1, ϱ), let θ∗

¯ϱ∈[0, θϱ)denote the unique

root of the function w5(·,¯

ζϱ,¯ϱ)given in (2.181). Then the candidate strategy

Xθ∗

¯ϱ,ζϱeκθ∗

¯ϱ,0in (2.66)with problem data (θ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0) is optimal and satisfies

¯ϱ∇↑

0Jθ∗

¯ϱ(Xθ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0) =0 and

dt (¯ϱ∇↑

tJθ∗

¯ϱ(Xθ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0))⏐⏐⏐⏐t↓0

>0.

(2.74)

In particular, (θ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0) ∈∂Rbuy.

2. Let ϱ= 1 be fixed. Then for all ζ∈(0, e−κθ12µ/κ]the assertion from 1.)

holds true with θ1=θ.

Observe that Lemma 2.3.13 1.) applies to all the triplets described in Proposi-

tion 2.3.27.

2.3.4.4 Negative risky asset holdings φ

Let us finalize our analysis by examining via the same approach as before all prob-

lem data (0, ζ, φ)∈Ssuch that φ < 0. In this case, the corresponding optimal

strategy ˆ

X0,ζ,φ is simply given by ˆ

X0,ζ,φ,↑

0=−φ > 0and ˆ

X0,ζ,φ,↓

0= 0; cf. Re-

mark 2.3.16. Thus, similar to (2.57) above, we consider for any θ≥0the problem

data (θ, ζeκθ, φ)∈Swith associated candidate strategy

Xθ,ζeκθ,φ

t≜(−φ1{θ}(t),0)(0 ≤t≤θ).(2.75)

That is, the strategy in (2.75) is inactive until time θand then evens with a single

block buy order its negative position in the risky asset φ < 0. As it turns out

below, only the case θ= 0 will be relevant for characterizing the last bit of the free

boundary ∂Rbuy in the state space S.

Proposition 2.3.28. Let φ < 0be arbitrary and set

ζ≜s4(φ)≜2µ−φ(2λ2+ηκ)

κ>0.(2.76)

1. For any ϑ > 0the optimal strategy ˆ

X0,¯

ζ,φ in (2.75)with θ= 0 and problem

data (0,¯

ζ, φ)∈Scan be extended to obtain an optimal strategy ˆ

Xϑ,ζ′,φ′∈Xd

with problem data (ϑ, ζ′, φ′)∈∂Rbuy such that

φˆ

Xϑ,ζ′,φ′

ϑ−=φand ζˆ

Xϑ,ζ′,φ′

ϑ−=¯

ζ. (2.77)

On [0, ϑ], this strategy as well as the pair (ζ′, φ′)are given as in Corol-

lary 2.3.21 in (2.53)and (2.54), respectively, with τ= 0,cevaluated at

(¯

ζ, φ)and ζ′>¯

ζ.

2 Optimal Investment with Transient Price Impact

2. For all ζ > ¯

ζthe candidate strategy Xθ,ζeκθ,φ in (2.75)is optimal for any

θ > 0and satisfies

∇↑

0Jθ(Xθ,ζeκθ,φ)>0.(2.78)

In particular, (θ, ζeκθ, φ)∈Rwait for all θ > 0.

Remark 2.3.29.1. Note that the optimal investment strategies ˆ

Xϑ,ζ′,φ′obtained

in Proposition 2.3.28 1.) are strategies that continuously buy the risky as-

set during the entire period [0, ϑ]. Therefore, they can be considered as the

counterparts to the sell-only strategies described in Proposition 2.3.17 in Sec-

tion 2.3.3. In contrast, regarding the characterization of the free boundary of

the buying-region ∂Rbuy, these strategies only contribute to a minor extent

to the description of the latter.

2. Analogously to Remark 2.3.19 1.) concerning the optimal sell-only strate-

gies from Proposition 2.3.17, observe that the optimal buy-only strategies

from Proposition 2.3.28 1.) unwind with a final block buy order the remain-

ing shares φ < 0at final time ϑwhich is in line with our convention from

Remark 2.3.16.

3. In fact, let us mention that letting φ= 0 in (2.76) above, the resulting optimal

strategy will coincide with the one from Corollary 2.3.26 in the case ϱ=−1.

2.3.5 Synopsis and numerical illustration

Let us discuss how the results from the preceding Sections 2.3.3 and 2.3.4 allow us

to compute explicitly the free boundaries of the selling region ∂Rsell and buying

region ∂Rbuy in the three-dimensional state space S. We are keeping the same

notations introduced before. For ease of presentation, we accompany our expla-

nations with a numerical example where the model parameters are simply chosen

κ= 1, η = 2, µ = 10, σ = 1, α = 1,

i.e., the Merton portfolio is given by φ0=µ/(ασ2) = 10.

Figure 2.1 shows the three-dimensional state space Swith time to maturity τ,

spread ζand number of shares φ. The blue plane represents the Merton plane M

from Definition 2.3.11. The upper red surface illustrates the boundary of the selling

region ∂Rsell which is obtained from equation (2.33) in Proposition 2.3.17, i.e.,

∂Rsell ={(τ, ζ, φ)∈S:φ=ϕ1(τ, ζ)}.

Consequently, due to Corollary 2.3.18 (recall also Lemma 2.3.13), we have

Rsell ={(τ, ζ, φ)∈S:φ > ϕ1(τ, ζ)}.

2.3 Case study: Illiquid Bachelier model with exponential utility

Figure 2.1: The three-dimensional state space Swith Merton plane M(blue),

boundary of the buying-region ∂Rbuy (colored in varying green) and

boundary of the selling-region ∂Rsell (red). The thicker green colored

curve separating the meshed and non-meshed parts of ∂Rbuy represents

the free boundary curve detected in Section 2.3.4.

The red free boundary ∂Rsell is completely traced out by the evolution in Sof the

corresponding optimally controlled state processes in the sense of Lemmas 2.3.14

and 2.3.15 whose dynamics satisfy the identity in (2.36) given by

φˆ

Xτ,ζ,φ

t=ϕ1(τ−t, ζ ˆ

Xτ,ζ,φ

t) (0 ≤t≤τ).

The associated optimal share holdings φˆ

Xτ,ζ,φ satisfy the ODE in (2.29), namely

¨φˆ

Xτ,ζ,φ

t=β2(φˆ

Xτ,ζ,φ

t−µ

ασ2)(0 < τ < t).

The lower green surface depicts the free boundary of the buying region ∂Rbuy

as characterized in Section 2.3.4. It decomposes into eight parts, all plotted in

different shades of green with and without mesh. The thicker green colored curve

separating the meshed and non-meshed parts of the buying boundary represents

the free boundary curve embedded in ∂Rbuy and identified in Propositions 2.3.20

1.), 2.3.23 1.) and 2.3.25 1.) via parametrizations in τ,θand ϱ, respectively,

namely

τ↦→ (τ+¯

θ, s1(τ)eκ¯

θ, ϕ1(τ, s1(τ))), θ ↦→ (θ, s2(θ)eκθ, ϕ2(θ)), ϱ ↦→ (θϱ,2µ/κ, 0).

These three different bits of the curve are plotted correspondingly in three different

shades of green. The corresponding three green colored meshed parts of the free

boundary ∂Rbuy are obtained by the evolution of the optimally controlled state

2 Optimal Investment with Transient Price Impact

processes (more precisely, by their buying parts) described in Corollary 2.3.21,

2.3.24 and 2.3.26, respectively, again in the sense of Lemmas 2.3.14 and 2.3.15.

The optimal share holdings as well as the corresponding spread of these buying

trajectories are fully characterized by the pair (ζ, φ)that they pass through on the

free boundary curve; cf. the representation of the optimal cumulative purchases

X↑in (2.53). As in case of the boundary of the selling region, the optimal share

holdings φˆ

Xsatisfy likewise during the buying period on [0, ϑ]the ODE

¨φˆ

t=β2(φˆ

t−µ

ασ2)(0 < t < ϑ)

with boundary conditions φˆ

ϑ=φand ζˆ

ϑ=ζgiven by the pair (ζ, φ)on the free

boundary curve. Analogously, the fourth lightest green meshed part corresponds

to the case φ < 0from Proposition 2.3.28 1.). The remaining four non-meshed

parts correspond, respectively,

1. to the triplets (τζ+θ∗, ζeκθ∗,¯φ)∈∂Rbuy with (τζ, ζ, ¯φ)∈∂Rsell obtained in

Proposition 2.3.20 2.) satisfying the relation

(λ2θ∗−λ2

κ(e−κθ∗+ 1))¯φ+1

2(e−κθ∗+2+eκθ∗)ζ+µ

κ(e−κθ∗+ 1)−µθ∗= 0;

2. to the triplets (θ∗, ζeκθ∗,¯φ)∈∂Rbuy obtained in Proposition 2.3.23 2.) sat-

isfying

(λ2¯φ−µ)θ∗+1

2ζ(eκθ∗+ 1)+1

2η¯φ(e−κθ∗+ 1)= 0

3. and to the triplets (θ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0) ∈∂Rbuy obtained in Proposition 2.3.27

satisfying

−µθ∗

¯ϱ+1

2¯

ζϱ(eκθ∗

¯ϱ+ ¯ϱ)= 0.

In particular, we observe the “plateau” on the free boundary ∂Rbuy of the buying-

region along the free boundary curve ϱ↦→ (θϱ,2µ/κ, 0) ∈∂Rbuy as described in

Proposition 2.3.27 which is induced by the dependence of the buy-subgradients on

ϱ= signϱ(0) ∈[−1,1].

As already discussed in Remark 2.3.12 above, Figure 2.1 illustrates that the

red boundary of the selling region ∂Rsell is above the Merton plane Mfor large

maturities τbut falls below Mfor small maturities and small spread values ζ.

In contrast, the green boundary of the buying region ∂Rbuy is always below the

Merton plane.

Figure 2.2 depicts the evolutions of some optimal share holdings (φˆ

Xτ,ζ,φ

t)0≤t≤τ

for different problem data (τ, ζ, φ)∈S. The evolutions of the corresponding opti-

mally controlled state processes (τ−t, ζ ˆ

Xτ,ζ,φ

t, φ ˆ

Xτ,ζ,φ

t)0≤t≤τin the state space S

are illustrated in Figure 2.3. For example, the red policy is an optimal liquidation

2.3 Case study: Illiquid Bachelier model with exponential utility

012345 τ

-5

Figure 2.2: Evolution of optimal share holdings for different initial problem data

(τ, ζ, φ)∈S. The dots represent the inital position in the risky asset.

The final position is always zero. The grey line represents the Merton

position φ0=µ/(ασ2) = 10.

Figure 2.3: Evolution of optimally controlled state processes embedded in the three-

dimensional state space Scorresponding to the optimal share holdings

from Figure 2.2. Dashed lines indicate waiting parts of the strategies

and the big dots represent the corresponding initial and final triplets

(τ, ζ, φ)and (0, ζ′,0), respectively, for some final spread value ζ′. In

particular, by our convention from Remark 2.3.16, all strategies unwind

non-zero positions in the end with an impulse trade.

2 Optimal Investment with Transient Price Impact

strategy as described in Corollary 2.3.18 above. The trajectory starts in Rsell, im-

mediately “jumps” with an initial block sell order on the free boundary of the selling

region ∂Rsell, continues selling steadily the risky asset until final time and then liq-

uidates the remaining shares with a single block sell order. The blue policy is an

optimal strategy obtained from Corollary 2.3.21 (together with Remark 2.3.22 4.)).

Since the initial problem data belongs to the waiting region ∂Rwait, it decomposes

into a waiting-, buying-, waiting- and selling part. The black policy is of buy-

and-hold type with initial and final block trades obtained from Proposition 2.3.23

2.) (together with Lemma 2.3.13 1.)). The pink policy does not trade at all and

unwinds at the end. Its initial problem data belongs to those described in Proposi-

tion 2.3.23 3.). The magenta policy is obtained from Proposition 2.3.27 1.) (again

together with Lemma 2.3.13 1.)). It evens with a single block buy order its initial

negative risky asset position, i.e., “jumps” to the “plateau”, and then does not do

anything until maturity.

To sum up, we observe that, depending on time to maturity τ, initial spread ζ

and initial share holdings φ, the optimal share holdings to the optimal tracking

problem in (2.13) from Proposition 2.3.2 in our illiquid Bachelier model exhibit a

rich phenomenology of possible trajectories.

2.4 Proofs

2.4.1 Proofs of Sections 2.1 and 2.2

We start with the computation of the dynamics of the liquidation wealth process

(Vt(X))t≥0defined in (2.4) and the associated liquidity costs (Lt(X))t≥0stated in

Lemma 2.1.2.

Proof of Lemma 2.1.2.To alleviate the notation, let us introduce the mid-

quote price process MX

t≜(AX

t+BX

t)/2for all t≥0with initial value MX

0−≜

(A0+B0)/2 = P0−. Applying integration by parts in (2.4) as in, e.g., Jacod and

Shiryaev (2003), Definition I.4.45, yields

dVt(X) = −1

2(ζX

t−+η∆X↑

t)dX↑

t−1

2(ζX

t−+η∆X↓

t)dX↓

t+φX

t−dMX

−ηφX

t−dφX

t−1

2(ζX

t−d|φX

t|+|φX

t−|dζX

t+d[|φX|, ζX]t),(2.79)

where we used the fact that [φX, MX] = η[φX, φX]/2by virtue of Jacod and

Shiryaev (2003), Theorem I.4.52. Moreover, note that Proposition I.4.49 a) in

Jacod and Shiryaev (2003) implies

[|φX|, ζX]t=∫[0,t]

∆ζX

sd|φX

s|(t≥0),(2.80)

since |φX|is predictable and ζXis of finite variation. Inserting (2.80), the spread

2.4 Proofs

dynamics (2.2) as well as the dynamics of the mid-quote

dMX

t=dPt+1

2ηdX↑

t−1

2ηdX↓

t(t≥0)

in (2.79) above yields

dVt(X) = φX

t−dPt−1

2ζX

td|φX

t|+1

2κ|φX

t−|ζX

t−dt

−1

2(ζX

t−+η∆X↑

t+ηφX

t−+η|φX

t−|)dX↑

−1

2(ζX

t−+η∆X↓

t−ηφX

t−+η|φX

t−|)dX↓

t(t≥0).

(2.81)

This motivates the definition of the liquidation cost functional Lt(X)via

Lt(X)≜L0−(X) + 1

2∫[0,t]

ζX

sd|φX

s|− 1

2κ∫[0,t]|φX

s−|ζX

s−ds

2∫[0,t](ζX

s−+η∆X↑

s+ηφX

s−+η|φX

s−|)dX↑

2∫[0,t](ζX

s−+η∆X↓

s−ηφX

s−+η|φX

s−|)dX↓

s(t≥0)

(2.82)

with L0−(X)≜ζ0|φX

0−|/2 + η(φX

0−)2/2. Using once more the spread dynamics in

(2.2) we can write

−1

2κ|φX

t−|ζX

t−dt =1

2|φX

t−|dζX

t−1

2η|φX

t|(dX↑

t+dX↓

t) (t≥0).

Inserting this expression in (2.82) gives us

Lt(X) = L0−(X) + 1

2∫[0,t]

(ζX

s−+η∆X↑

s)dX↑

2∫[0,t]

(ζX

s−+η∆X↓

s)dX↓

s+1

2∫[0,t]

ζX

sd|φX

2∫[0,t]|φX

s−|dζX

s+1

2η∫[0,t]

φX

s−dφX

s(t≥0).

(2.83)

Again, integration by parts as in Jacod and Shiryaev (2003), Definition I.4.45,

together with the identity in (2.80) allows us to write

2∫[0,t]|φX

s−|dζX

s=1

2|φX

t|ζX

t−1

2|φX

0−|ζX

0−−1

2∫[0,t]

ζX

sd|φX

s|,(2.84)

2η∫[0,t]

φX

s−dφX

s=1

4η((φX

t)2−(φX

0−)2−[φX, φX]t).(2.85)

Plugging back (2.84) and (2.85) into (2.83), using the definition of L0−(X)as well

as the fact that

[X↑,↓, X↑,↓]t=∫[0,t]

∆X↑,↓

sdX↑,↓

s(t≥0),

2 Optimal Investment with Transient Price Impact

cf. Proposition I.4.49 a) in Jacod and Shiryaev (2003), finally yields

Lt(X) = 1

2|φX

t|ζX

t+η

4((φX

t)2+ (φX

0−)2)

+η

4([X↑, X↑]t+ [X↓, X↓]t+ 2[X↑, X↓]t)

2∫[0,t]

ζX

s−(dX↑

s+dX↓

s) (t≥0).

(2.86)

Next, using the explicit representation of the spread ζXin (2.3) and defining the

process Yt≜∫[0,t]eκs(dX↑

s+dX↓

s)for all t≥0allows us to write

∫[0,t]

ζX

s−(dX↑

s+dX↓

=∫[0,t]

e−κsζ0(dX↑

s+dX↓

s) + ∫[0,t]

ηe−κsYs−(dX↑

s+dX↓

=∫[0,t]

e−κsζ0(dX↑

s+dX↓

s) + ∫[0,t]

ηe−2κsYs−dYs

=∫[0,t]

e−κsζ0(dX↑

s+dX↓

s) + ∫[0,t]

ηe−2κs 1

2(dY 2

s−d[Y, Y ]s)

=∫[0,t]

e−κsζ0(dX↑

s+dX↓

s) + η

2e−2κtY2

+κη ∫t

e−2κsY2

s−ds −η

2∫[0,t]

e−2κsd[Y, Y ]s.(2.87)

Once more due to Jacod and Shiryaev (2003), Proposition I.4.49, observe that we

have

d[Y, Y ]s=eκs(∆X↑

s+ ∆X↓

s)dYs

=e2κs (d[X↑, X↑]s+d[X↓, X↓]s+ 2d[X↑, X↓]s).(2.88)

Moreover, it holds that

e−κtYt=1

η(ζX

t−e−κtζ0)(t≥0).(2.89)

Thus, using (2.89) and (2.88) in (2.87) and plugging back the resulting expression

into (2.86) yields the desired expression of the liquidity cost functional in (2.6).

Finally, with the obtained expression for the liquidity costs one can easily observe

that the functional Lt(X)is convex in Xfor each t≥0. Moreover, using the lower

estimate ζX

t−e−κtζ0≥ηe−κt(X↑

t+X↓

t)for all t≥0, we obtain the lower bound

of Lt(X)as claimed in (2.7).

In order to apply Lemma 2.2.1 in our setting in the proof of Theorem 2.2.3 above,

we need the following lemma.

Lemma 2.4.1. For the level-set L0≜{X∈X:Eu(VT(X)) ≥Eu(VT(0))},

conv({X↑

T+X↓

T:X∈L0})is L0(Ω,F,P)-bounded.

2.4 Proofs

Proof. First, note that due to convexity of the liquidity cost functional LT(X)in

X∈Xby virtue of Lemma 2.1.2 as well as concavity and monotonicity of the

utility function u, the level-set L0is a convex set. Indeed, for every X, Y ∈L0

and λ∈[0,1] we obtain for Z:= λX + (1 −λ)Ythe lower bound

E[u(VT(Z))] ≥E[u(λVT(X)) + (1 −λ)u(VT(Y))] ≥E[u(VT(0))].

As a consequence, it holds that conv({X↑

T+X↓

T:X∈L0}) = {X↑

T+X↓

T:X∈L0}.

Next, observe that for any X∈Xthe liquidation wealth VT(X)as given in (2.5)

can be bounded from above by

VT(X) = V0−(X) + L0−(X) + ∫T

φX

tdPt−LT(X)

=ξX

0−+φX

TPT−∫[0,T ]

PtdφX

t−LT(X)

≤ξX

0−+ 2(φX

0−+X↑

T+X↓

T)P∗

T−c(X↑

T+X↓

T)2

=ξX

0−+1

c(P∗

T)2−(√c(X↑

T+X↓

T)−1

√cP∗

T)2+ 2φX

0−P∗

T(2.90)

with P∗

T≜max0≤s≤T|Ps|, where we used integration by parts, the fact that the

semimartingale (Pt)t≥0is continuous and the lower bound LT(X)≥c(X↑

T+X↓

T)2

from Lemma 2.1.2 for some constant c > 0. Henceforth, to alleviate the presen-

tation, let us assume without loss of generality that ξX

0−=φX

0−= 0 as well as

u(0) = 0. Due to the upper bound in (2.90), we obtain for all X∈L0the estimate

E[u(VT(0))] ≤E[u(VT(X))]

≤E[u(1

c(P∗

T)2−(√c(X↑

T+X↓

T)−1

√cP∗

T)2)].

Hence, together with the fact that uis bounded from above, it must hold for the

negative part that

sup

X∈L0

E⎡

⎣u(1

c(P∗

T)2−(√c(X↑

T+X↓

T)−1

√cP∗

T)2)−⎤

⎦<∞.(2.91)

Moreover, since u∈C1(R)is strictly concave and increasing which yields u(z)≤

u(0) + u′(0)z=u′(0)zand thus u(z)−≥u′(0)(−z)+for all z∈R, we obtain

sup

X∈L0

E⎡

⎣((√c(X↑

T+X↓

T)−1

√cP∗

T)2

−1

c(P∗

T)2)+⎤

⎦<∞.(2.92)

Finally, observe that the L1(Ω,F,P)-boundedness in (2.92) implies that the set

{X↑

T+X↓

T:X∈L0}is bounded in L0(Ω,F,P).

2 Optimal Investment with Transient Price Impact

The last ingredient for the proof of Theorem 2.2.3 is the lower semi-continuity

of the liquidation wealth VT(X)in X.

Proof of Lemma 2.2.2.We fix ω∈Ω. Note that pointwise convergence of Xn

to Xon Ω×[0, T]implies weak convergence in the sense that Xn(ω)w

−→ X(ω)on

[0, T]. Consequently, we obtain that ζXn

t(ω)→ζX

t(ω)for t=Tand all t∈[0, T)

such that ∆X↑

t(ω) = ∆X↓

t(ω) = 0; cf. the representation of the spread in (2.3). In

other words, it holds that ζXn

·(ω)→ζX

·(ω)dt-a.e. on [0, T]because the number of

jumps of X↑(ω),X↓(ω)is countable. Thus, an application of Fatou’s lemma yields

lim inf

n→∞ ∫t

0(ζXn

s(ω)−e−κsζ0)2ds ≥∫t

0(ζX

s(ω)−e−κsζ0)2ds.

Moreover, we obviously have that φXn

T(ω)→φX

T(ω). Hence, referring to the rep-

resentation of the liquidity costs LT(Xn)in (2.6), we can conclude that

lim inf

n→∞ LT(Xn(ω)) ≥LT(X(ω))

and thus lim supn→∞(−LT(Xn(ω))) ≤ −LT(X(ω)). Next, concerning the stochas-

tic integral of φXnwith respect to the continuous semimartingale Pin the liqui-

dation wealth VT(Xn)in (2.5), we obtain, after applying integration by parts, the

expression

∫T

φX

tdPt=φX

TPT−φX

0−P0−−∫[0,T ]

Ps(dX↑

s−dX↓

= lim

n→∞(φXn

TPT−φXn

0−P0−−∫[0,T ]

Ps(dX↑,n

s−dX↓,n

s))

= lim

n→∞∫T

φXn

tdPtfor all ω∈Ω,

where we again used weak convergence of Xn(ω)w

−→ X(ω)on [0, T]for all ω∈Ω

and the continuity of P. In summary, we obtain lim supn→∞ VT(Xn)≤VT(X)

pointwise for all ω∈Ωas desired.

2.4.2 Proofs of Lemma 2.3.6 and Lemma 2.3.8

Next, let us compute the infinite-dimensional subgradients (2.15) and (2.16) of the

convex cost functional JT(·)on Xdgiven in (2.13).

Proof of Lemma 2.3.6.Let us define the deviation functional

DT(X)≜ασ2

2∫T

0(φX

t−µ

ασ2)2

dt (2.93)

on Xd. Then, the convex cost functional JT(·)in (2.13) is given by

JT(X) = LT(X) + DT(X).

2.4 Proofs

We will proceed in three steps.

Step 1: Let us start with the computation of the subgradients of the liquidity cost

functional LT(·)given in (2.6). Observe that for any X, Y ∈Xdwith φY

0−=φX

0−

and any ε∈(0,1] we obtain

LT(εY + (1 −ε)X)−LT(X)

=η

4|εφY

T+ (1 −ε)φX

T|2−|φX

T|2

ε+1

2ζX

T|εφY

T+ (1 −ε)φX

T|−|φX

2(ζY

T−ζX

T)(|εφY

T+ (1 −ε)φX

T|+1

η(ζX

T−e−κT ζ0))

+κ

η∫T

(ζX

t−e−κtζ0)(ζY

t−ζX

t)dt

2∫[0,T ]

e−κtζ0(dY ↑

t+dY ↓

t−dX↑

t−dX↓

+ε(1

4η(ζY

T−ζX

T)2+κ

2η∫T

(ζY

t−ζX

t)2dt).(2.94)

For the first two terms in (2.94) we have the lower bound

|εφY

T+ (1 −ε)φX

T|2−|φX

T|2

ε≥2φX

T(φY

T−φX

T)(2.95)

and

|εφY

T+ (1 −ε)φX

T|−|φX

ε≥signϱ(φX

T)(φY

T−φX

T),(2.96)

respectively. Recall that we denote by x↦→ signϱ(x)the subgradient of the function

x↦→ |x|with signϱ(0) = ϱ∈[−1,1]; cf. Remark 2.3.5. Plugging back (2.95) and

(2.96) into (2.94) and passing to the limit ε↓0yields

lim

ε↓0

LT(εY + (1 −ε)X)−LT(X)

≥(η

2φX

T+1

2signϱ(φX

T)ζX

T)(φY

T−φX

2(|φX

T|+1

η(ζX

T−e−κT ζ0))(ζY

T−ζX

+κ

η∫T

(ζX

t−e−κtζ0)(ζY

t−ζX

t)dt

2∫[0,T ]

ζ0e−κt(dY ↑

t−dX↑

t+dY ↓

t−dX↓

t).(2.97)

Next, let us express every term in (2.97) as an integral with respect to either Y↑−X↑

or Y↓−X↓. Using the expression in (2.3) for the spread ζYand ζX, respectively,

2 Optimal Investment with Transient Price Impact

as well as Fubini’s Theorem, we can rewrite the third term in (2.97) as

η∫T

(ζX

t−e−κtζ0)(ζY

t−ζX

t)dt

=κ∫[0,T ](∫T

(ζX

t−e−κtζ0)e−κ(t−s)dt)(dY ↑

s−dX↑

+κ∫[0,T ](∫T

(ζX

t−e−κtζ0)e−κ(t−s)dt)(dY ↓

s−dX↓

s).(2.98)

Moreover, using

φY

T−φX

T=∫[0,T ]

(dY ↑

s−dX↑

s)−∫[0,T ]

(dY ↓

s−dX↓

s)(2.99)

as well as

ζY

T−ζX

T=∫[0,T ]

ηe−κ(T−s)(dY ↑

s−dX↑

s) + ∫[0,T ]

ηe−κ(T−s)(dY ↓

s−dX↓

allows us to finally write (2.97) as

lim

ε↓0

LT(εY + (1 −ε)X)−LT(X)

≥∫[0,T ]

ϱ∇↑

sLT(X)(dY ↑

s−dX↑

s) + ∫[0,T ]

ϱ∇↓

sLT(X)(dY ↓

s−dX↓

s),(2.100)

where we set

ϱ∇↑,↓

sLT(X)≜κ∫T

(ζX

t−e−κtζ0)e−κ(t−s)dt

2(η|φX

T|+ζX

T−e−κT ζ0)e−κ(T−s)

2ζ0e−κs ±η

2φX

T±1

2signϱ(φX

T)ζX

=κ∫T

e−κ(t−s)ζX

tdt +1

2(η|φX

T|+ζX

T)e−κ(T−s)

±η

2φX

T±1

2signϱ(φX

T)ζX

T(0 ≤s≤T).(2.101)

Step 2: Now, let us compute the subgradients of the deviation functional DT(·)

defined in (2.93). Again, for any X, Y ∈Xdwith φY

0−=φX

0−and any ε∈(0,1] we

obtain

DT(εY + (1 −ε)X)−DT(X)

=ασ2∫T

0(φX

t−µ

ασ2)(φY

t−φX

t)dt +εασ2

2∫T

(φY

t−φX

t)2dt

2.4 Proofs

and hence, together with (2.99) and Fubini’s Theorem, we arrive at

lim

ε↓0

DT(εY + (1 −ε)X)−DT(X)

=ασ2∫T

0(φX

t−µ

ασ2)(φY

t−φX

t)dt

=ασ2∫[0,T ](∫T

s(φX

t−µ

ασ2)dt)(dY ↑

s−dX↑

+ασ2∫[0,T ](∫T

s(µ

ασ2−φX

t)dt)(dY ↓

s−dX↓

s).

Consequently, we can write

lim

ε↓0

DT(εY + (1 −ε)X)−DT(X)

=∫[0,T ]∇↑

sDT(X)(dY ↑

s−dX↑

s) + ∫[0,T ]∇↓

sDT(X)(dY ↓

s−dX↓

s),(2.102)

where we set

∇↑,↓

sDT(X)≜±ασ2∫T

s(φX

t−µ

ασ2)dt (0 ≤s≤T).(2.103)

Step 3: Finally, regarding the convex cost functional JT(·)we obtain for any

X, Y ∈Xdwith φY

0−=φX

0−and any ε∈(0,1] the lower bound

JT(Y)−JT(X)≥JT(εY + (1 −ε)X)−JT(X)

=LT(εY + (1 −ε)X)−LT(X)

+DT(εY + (1 −ε)X)−DT(X)

ε.

Passing to the limit ε↓0yields together with (2.100) and (2.102)

JT(εY + (1 −ε)X)−JT(X)

≥∫[0,T ]

(ϱ∇↑

sLT(X) + ∇↑

sDT(X))(dY ↑

s−dX↑

+∫[0,T ]

(ϱ∇↓

sLT(X) + ∇↓

sDT(X))(dY ↓

s−dX↓

s),

where we note that

ϱ∇↑,↓

sJT(X) = ϱ∇↑,↓

sLT(X) + ∇↑,↓

sDT(X) (0 ≤s≤T)

as desired.

2 Optimal Investment with Transient Price Impact

Proof of Lemma 2.3.8.Let X= (X↑, X↓)∈Xd,X= (0,0), be a strategy such

that ∇↑

tJT(X) = 0 for some t∈[0, T]. Then, in view of the definition of ∇↑

tJT(X)

in (2.15) it holds that

−η

2φX

T−1

2sign(φX

T)ζX

=∫T

t(κe−κ(u−t)ζX

u+ασ2(φX

u−µ

ασ2))du +1

2(η|φX

T|+ζX

T)e−κ(T−t).

Using this identity in the definition of ∇↓

tJT(X)in (2.16) yields

∇↓

tJT(X) =∫T

t(κe−κ(u−t)ζX

u+ασ2(µ

ασ2−φX

u))du

2(η|φX

T|+ζX

T)e−κ(T−t)

−η

2φX

T−1

2sign(φX

T)ζX

=2∫T

κe−κ(u−t)ζX

udu +(η|φX

T|+ζX

T)e−κ(T−t)>0

because X= (0,0). The same computations apply in the case where the roles of ↑

and ↓are interchanged.

2.4.3 Proofs of Section 2.3.2

Proof of Lemma 2.3.13.We only verify the assertion in 1.). The assertion

in 2.) follows by similar arguments. Therefore, let (τ, ζ, φ)∈Rbuy,τ > 0,

be some arbitrary problem data with corresponding optimal strategy ˆ

Xτ,ζ,φ =

(ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)∈Xdsatisfying ϱ∇↑

0Jτ(ˆ

Xτ,ζ,φ) = 0 and ˆ

Xτ,ζ,φ,↑

0= 0 for some

ϱ∈[−1,1]. Moreover, let 0< x ≤ζ/η be arbitrary but fixed. Then, we have

(τ, ζ −ηx, φ)∈S. In addition, observe that the candidate strategy ˆ

Xτ,ζ−ηx,φ−xin

(2.26) obviously belongs to the set Xdand satisfies for all t∈[0, τ]the identities

φˆ

Xτ,ζ−ηx,φ−x

t=φ−x+x+ˆ

Xτ,ζ,φ,↑−ˆ

Xτ,ζ,φ,↓=φˆ

Xτ,ζ,φ

as well as

ζˆ

Xτ,ζ−ηx,φ−x

t=e−κt(ζ−ηx) + ηxe−κt +∫[0,t]

eκu(dˆ

Xτ,ζ,φ,↑

u+dˆ

Xτ,ζ,φ,↓

u) = ζˆ

Xτ,ζ,φ

cf. the spread dynamics in (2.3). Consequently, appealing to the definition of the

subgradients in (2.15) and (2.16), it holds that

ϱ∇↑,↓

tJτ(ˆ

Xτ,ζ−ηx,φ−x) = ϱ∇↑,↓

tJτ(ˆ

Xτ,ζ,φ) (0 ≤t≤τ).

Thus, we have ϱ∇↑

0Jτ(ˆ

Xτ,ζ−ηx,φ−x) = 0 by assumption. In particular, ˆ

Xτ,ζ−ηx,φ−x

is optimal in view of the first order conditions in Proposition 2.3.7 and the fact

that ˆ

Xτ,ζ,φ is optimal.

2.4 Proofs

Proof of Lemma 2.3.14.Let (τ, ζ, φ)∈Swith τ > 0be some arbitrary

data with optimal strategy ˆ

X≜ˆ

Xτ,ζ,φ ∈Xd. We only consider the case where

[0, v]⊂ {dˆ

X↑>0} ⊂ {ϱ∇↑Jτ(ˆ

X↑) = 0}for some ϱand 0< v ≤τ(the other case

when interchanging ↑by ↓can be treated analogously). We start with arguing that

X↑is absolutely continuous on (0, v). By assumption, it holds that ϱ∇↑

tJτ(ˆ

X) = 0

for all t∈[0, v](and some suitable ϱ), i.e., using the definition in (2.15), we have

ϱ∇↑

tJτ(ˆ

X) =∫τ

t(κe−κ(u−t)ζˆ

u+ασ2(φˆ

u−µ

ασ2))du

2(η|φˆ

τ|+ζˆ

τ)e−κ(τ−t)

+η

2φˆ

τ+1

2signϱ(φˆ

τ)ζˆ

τ= 0 for all t∈[0, v].(2.104)

Observe that the mapping t↦→ ϱ∇↑

tJτ(ˆ

X)is differentiable. Thus, it follows from

(2.104) that we also have

dt (ϱ∇↑

tJτ(ˆ

X))= 0 for all t∈(0, v).

Consequently, differentiating equation (2.104) with respect to tand multiplying by

e−κt yields

0 =∫τ

κ2e−κuζˆ

udu −κe−κtζˆ

t+µe−κt

−ασ2φX

te−κt +κ

2e−κτ (η|φˆ

τ|+ζˆ

τ)on (0, v).(2.105)

Using integration by parts in (2.105) and multiplying with eκt results in the equa-

tion

−κdζ ˆ

t−κµdt +ασ2κφ ˆ

tdt −ασ2dφ ˆ

t= 0 on (0, v).(2.106)

Plugging the spread dynamics from (2.2) into (2.106) and using the fact that [0, v]⊂

{dˆ

X↑>0}\{dˆ

X↓>0}due to Lemma 2.3.8, we obtain

−κηd ˆ

X↑

t+κ2ζˆ

tdt −κµdt +κασ2φˆ

tdt −ασ2dˆ

X↑

t= 0 on (0, v)

and thus the relation

dˆ

X↑

t=κ2ζˆ

t+κασ2φˆ

t−κµ

κη +ασ2dt on (0, v)(2.107)

which implies that the process ˆ

X↑necessarily has to be absolutely continuous on

(0, v). As a consequence, for any problem data (τ−t, ζ ˆ

t, φ ˆ

t)with 0< t < v,

the associated optimal strategy ˆ

X′≜ˆ

Xτ−t,ζ ˆ

t,φ ˆ

tsatisfies ˆ

X′,↑

0= 0, that is, (2.28)

holds true. Indeed, by the dynamic programming principle from Remark 2.3.9, it

holds that ˆ

X′,↑

u=ˆ

X↑

t+u−ˆ

X↑

t−(0 ≤u≤τ−t).

2 Optimal Investment with Transient Price Impact

Hence, ˆ

X′,↑

0=ˆ

X↑

t−ˆ

X↑

t−= 0 due to the continuity of ˆ

X↑in t∈(0, v).

Proof of Lemma 2.3.15.Let us prove the claim in the case where [u, v]⊂ {dˆ

X↑>

0} ⊂ {ϱ∇↑Jτ(ˆ

X) = 0}for some ϱand some optimal strategy ˆ

X= ( ˆ

X↑, X↓).

The same reasoning applies in the case [u, v]⊂ {dˆ

X↓>0} ⊂ {ϱ∇↓Jτ(ˆ

X) =

0for some ϱ}. In view of the proof of Lemma 2.3.14 above, in particular the identity

in (2.107), we know that φˆ

Xis absolutely continuous on (u, v)with density

˙φˆ

t=κ2ζˆ

t+κασ2φˆ

t−κµ

κη +ασ2on (u, v).(2.108)

Hence, using once more the spread dynamics from (2.2) and the fact that [u, v]⊂

{dˆ

X↑>0}\{dˆ

X↓>0}due to Lemma 2.3.8, it follows from integration by parts

that

(κη +ασ2)d˙φˆ

t=κ2dζ ˆ

t+κασ2dφ ˆ

=−κ3ζˆ

tdt +κ(κη +ασ2) ˙φˆ

tdt

=κ2ασ2(φˆ

t−µ

ασ2)dt on (u, v)

and thus the second order ODE

¨φˆ

t=κ2ασ2

κη +ασ2(φˆ

t−µ

ασ2)on (u, v)

as desired.

2.4.4 Proofs of Section 2.3.3

Lemma 2.4.2. The functions C(τ)and D(τ)defined in (2.32)satisfy C(τ)>0

and 1> D(τ)>0for all τ≥0. In particular, the function ϕ1defined in (2.33)

satisfies ϕ1(τ, ζ)>0for all τ≥0,ζ≥0. Moreover, we have C(0) = λ/(2λ2+κη)

and D(0) = 2λ2/(2λ2+κη).

Proof. Concerning the function C(τ), we note that C(τ)>0if and only if e−βτ γ−+

eβτ γ+>0, or, equivalently, if and only if 2βτ > log(−γ−/γ+)(observe that γ−<0

and γ+>0). Since β > 0, this is satisfied if −γ−/γ+<1, or, equivalently, if

−γ−< γ+, which obviously holds true. Finally, we can compute

C(0) = γ−+γ+

γ2

−+γ2

=λ

2λ2+κη.

Concerning the function D(τ), we note that γ−γ+=−κη and hence

1> D(τ) = 1 −2κη

e−βτ γ2

−+eβτ γ2

=(e−βτ/2γ−+eβτ/2γ+)2

e−βτ γ2

−+eβτ γ2

>0.

2.4 Proofs

Moreover, it holds that

D(0) = 1 −2κη

γ2

−+γ2

=2λ2

2λ2+κη.

Finally, note that C(τ)>0and D(τ)>0implies ϕ1(τ, ζ)>0for all τ≥0,

ζ≥0.

Proof of Proposition 2.3.17.Let (τ, ζ, φ)∈Ssatisfy (2.33). We will proceed in

four steps. In the first two steps, we argue that the strategy ˆ

Xτ,ζ,φ stated in Propo-

sition 2.3.17 satisfies ˆ

Xτ,ζ,φ,↓

0=ℓ0(τ, ζ, φ) = 0 and belongs to the set Xd. Then, we

will show that the controlled system (τ−t, ζ ˆ

Xτ,ζ,φ

t, φ ˆ

Xτ,ζ,φ

t)0≤t≤τsatisfies property

(2.36) and that φˆ

Xτ,ζ,φ solves the ODE in (2.29). Finally, we argue that ˆ

Xτ,ζ,φ is

optimal by using the first order optimality conditions from Proposition 2.3.7.

1.) We start with claiming that ˆ

Xτ,ζ,φ,↓

0=ℓ0(τ, ζ, φ) = 0. To show this, first

observe that the denominator of the functions c+,c−defined in (2.35) is strictly

positiv. Indeed, we have eβτ γ2

+−e−βτ γ2

−>0if and only if 2βτ > log((γ−/γ+)2).

Since β > 0, this is satisfied if (γ−/γ+)2<1, or, equivalently, if |γ−|< γ+.

This is indeed the case because γ−<0and hence |γ−|=−γ−=−λ+√κη +λ2<

λ+√κη +λ2=γ+. Now, reducing ˆ

Xτ,ζ,φ,↓

0=ℓ0(τ, ζ, φ) = c+(τ, ζ, ϕ)+c−(τ, ζ, ϕ)+

φ−µ/λ2to the common (strictly positive) denominator of c±will give us in the

resulting numerator the expression

κ(e−βτ γ−(ηµ −λ2(ζ+ηφ)) + ηµγ+)+κ(eβτ γ+(ηµ −λ2(ζ+ηφ)) + ηµγ−)

+φλ2√κη +λ2(eβτ γ2

+−e−βτ γ2

−)−µ√κη +λ2(eβτ γ2

+−e−βτ γ2

−)

=φλ2(−ηκe−βτ γ−−ηκeβτ γ++eβτ √κη +λ2γ2

+−e−βτ √κη +λ2γ2

−)

−ζκλ2(e−βτ γ−+eβτ γ+)−µ(−ηκe−βτ γ−−ηκγ+−ηκeβτ γ+−ηκγ−

+eβτ √κη +λ2γ2

+−e−βτ √κη +λ2γ2

−)

and hence, recalling γ±=λ±√κη +λ2,

φλ3(e−βτ γ2

−+eβτ γ2

+)−ζκλ2(e−βτ γ−+eβτ γ+)

−µλ (e−βτ γ2

−+eβτ γ2

++ 2γ−γ+)

which vanishes if and only if

φλ2(e−βτ γ2

−+eβτ γ2

=κλζ (e−βτ γ−+eβτ γ+)+µ(e−βτ γ2

−+eβτ γ2

++ 2γ−γ+).

But, in view of the definition of C(τ)and D(τ)in (2.32) and the fact that γ−γ+=

−κη, this is equivalent to the assumption in (2.33), i.e., φ=ϕ1(τ, ζ).

2 Optimal Investment with Transient Price Impact

2.) Next, let us show that ˆ

Xτ,ζ,φ,↓

t=ℓt(τ, ζ, φ)is nondecreasing. Observe that

it suffices to prove that c+(τ, ζ, φ)≥c−(τ, ζ, φ). A direct computation yields that

c+(τ, ζ, φ)≥c−(τ, ζ, φ)if and only if

ζ+ηφ ≥µη

λ2(1−2√κη +λ2

eβτ γ+−e−βτ γ−).(2.109)

We will argue that the assumption in (2.33) implies the inequality in (2.109). In-

deed, assumption (2.33) yields that

φ≥µ

λ2D(τ) = µ

λ2(1−2κη

e−βτ γ2

−+eβτ γ2

+)(2.110)

because C(τ)>0(cf. Lemma 2.4.2). Moreover, it holds that the inequality

κη

e−βτ γ2

−+eβτ γ2

+≤√κη +λ2

eβτ γ+−e−βτ γ−

(2.111)

is equivalent to the inequality eβτ γ2

+≥e−βτ γ2

−which is always satisfied as we have

argued above in 1.). Indeed, (2.111) is equivalent to

eβτ (κηγ+−√κη +λ2γ2

+)≤e−βτ (γ2

−√κη +λ2+κηγ−)

and hence −λeβτ γ2

+≤ −λe−βτ γ2

−. As a consequence, we can deduce from ζ≥0,

(2.110) and (2.111) that

ζ+ηφ ≥ηφ ≥µη

λ2(1−2κη

e−βτ γ2

−+eβτ γ2

≥µη

λ2(1−2√κη +λ2

eβτ γ+−e−βτ γ−)

as desired in (2.109). In summary, steps 1.) and 2.) yield that ˆ

Xτ,ζ,φ ∈Xd.

3.) Now, let us show that the controlled triplet (τ−t, ζ ˆ

Xτ,ζ,φ

t, φ ˆ

Xτ,ζ,φ

t)0≤t≤τalways

satisfies (2.36), i.e.,

φˆ

Xτ,ζ,φ

t−µ

λ2D(τ−t) + κ

λζˆ

Xτ,ζ,φ

tC(τ−t) = 0 (0 ≤t≤τ).(2.112)

For every t∈[0, τ]the stock holdings of our candidate are given by

φˆ

Xτ,ζ,φ

t=φ−ˆ

Xτ,ζ,φ,↓

t=−c+(τ, ζ, φ)eβt −c−(τ, ζ, φ)e−βt +µ

λ2.(2.113)

Concerning the spread, due to the dynamics in (2.3), it holds that

ζˆ

Xτ,ζ,φ

t=ζe−κt +ηe−κt ∫[0,t]

eκsdˆ

Xτ,ζ,φ,↓

=ζe−κt +c+(τ, ζ, φ)ηβ

κ+β(eβt −e−κt)

−c−(τ, ζ, φ)ηβ

κ−β(e−βt −e−κt) (0 ≤t≤τ).(2.114)

2.4 Proofs

Inserting (2.113) and (2.114) into the left hand side of (2.112) and reducing the

resulting expression to the common (strictly positive) denominator e−β(τ−t)γ2

−+

eβ(τ−t)γ2

+, we obtain in the numerator the expression

(−c+eβt −c−e−βt)(e−β(τ−t)γ2

−+eβ(τ−t)γ2

+) + 2κηµ

−κ

λ(e−β(τ−t)γ−+eβ(τ−t)γ+)ζˆ

Xτ,ζ,φ

=e2βtγ−c+e−βτ {−γ−−κ

ηβ

κ+β}

+e−2βtγ+c−eβτ {−γ++κ

ηβ

κ−β}(2.115)

+e(β−κ)tγ−

λe−βτ {−ζ+c+ηβ

κ+β−c−ηβ

κ−β}

+e(−β−κ)tγ+

λeβτ {−ζ+c+ηβ

κ+β−c−ηβ

κ−β}

−c+γ2

+eβτ −c−γ2

−e−βτ +2κηµ

λ2+γ−

c−ηβ

κ−βe−βτ −γ+

c+ηβ

κ+βeβτ ,

where we skipped the arguments in the functions c+(τ, ζ, φ)and c−(τ, ζ, φ). Now,

a simple but very tedious computation shows that all expressions in the curly

brackets in (2.115) above as well as the term in the final line actually vanish.

Thus, it follows that (2.112) holds true as desired. In particular, we have that

φˆ

Xτ,ζ,φ

t=ϕ1(τ−t, ζ ˆ

Xτ,ζ,φ

t)>0for all t∈[0, τ]by virtue of Lemma 2.4.2. Moreover,

observe that the share holdings in (2.113) clearly satisfy the ODE dynamics in (2.29)

on (0, τ).

4.) Finally, let us show that the strategy ˆ

Xτ,ζ,φ from Proposition 2.3.17 is

optimal. By virtue of the first order optimality conditions in Proposition 2.3.7, we

have to show that ˆ

Xτ,ζ,φ satisfies

∇↓

tJτ(ˆ

Xτ,ζ,φ) = 0 and ∇↑

tJτ(ˆ

Xτ,ζ,φ)≥0for all 0≤t≤τ. (2.116)

Hence, let us compute all the objects needed in the sell-subgradient ∇↓

tJτ(ˆ

Xτ,ζ,φ)

given in (2.16), i.e.,

∇↓

tJτ(ˆ

Xτ,ζ,φ) = κeκt ∫τ

e−κuζˆ

Xτ,ζ,φ

udu −λ2∫τ

φˆ

Xτ,ζ,φ

udu +µ(τ−t)

2(η|φˆ

Xτ,ζ,φ

τ|+ζˆ

Xτ,ζ,φ

τ)e−κ(τ−t)

−η

2φˆ

Xτ,ζ,φ

τ−1

2ζˆ

Xτ,ζ,φ

τ(0 ≤t≤τ).

(2.117)

Note that property (2.36) implies φˆ

Xτ,ζ,φ

τ>0, see also Remark 2.3.19. First, using

the spread ζˆ

Xτ,ζ,φ

t=ζe−κt +ηe−κt ∫[0,t]eκsdˆ

Xτ,ζ,φ,↓

s, we obtain after an application

2 Optimal Investment with Transient Price Impact

of Fubini’s Theorem

∫τ

e−κuζˆ

Xτ,ζ,φ

udu (2.118)

=−1

2κζ(e−2κτ −e−2κt) + η∫[0,τ]

eκs (∫τ

max{s,t}

e−2κudu)dˆ

Xτ,ζ,φ,↓

=−1

2κζ(e−2κτ −e−2κt)−η

2κ(e−2κτ −e−2κt)∫[0,t]

eκsdˆ

Xτ,ζ,φ,↓

−η

2κe−2κτ ∫(t,τ]

eκsdˆ

Xτ,ζ,φ,↓

s+η

2κe−2κτ ∫(t,τ]

e−κsdˆ

Xτ,ζ,φ,↓

Moreover, using the explicit representation ˆ

Xτ,ζ,φ,↓

t=ℓt(τ, ζ, φ)with ℓt(·)defined

in (2.34) we get

∫[0,t]

eκsdˆ

Xτ,ζ,φ,↓

s=c+β

κ+β(e(κ+β)t−1) −c−β

κ−β(e(κ−β)t−1),(2.119)

∫(t,τ]

eκsdˆ

Xτ,ζ,φ,↓

s=c+β

κ+β(e(κ+β)τ−e(κ+β)t)−c−β

κ−β(e(κ−β)τ−e(κ−β)t),

∫(t,τ]

e−κsdˆ

Xτ,ζ,φ,↓

s=c+β

β−κ(e(β−κ)τ−e(β−κ)t) + c−β

κ+β(e−(κ+β)τ−e−(κ+β)t),

where we skipped once more the arguments in the functions c+(τ, ζ, φ)and c−(τ, ζ, φ).

Hence, for the first integral in (2.117) we obtain

κeκt ∫τ

e−κuζˆ

Xτ,ζ,φ

udu

=eκt{−1

2ζe−2κτ −η

2e−2κτ (−c+β

κ+β+c−β

κ−β)

−η

2e−2κτ (c+β

κ+βe(κ+β)τ−c−β

κ−βe(κ−β)τ)

+η

c+β

β−κe(β−κ)τ+η

c−β

κ+βe−(κ+β)τ}

+e−κt {1

2ζ−η

c+β

κ+β+η

c−β

κ−β}+eβt {η

c+β

κ+β−η

c+β

β−κ}

+e−βt {−η

c−β

κ−β−η

c−β

κ+β}.(2.120)

Next, using the representation of φˆ

Xτ,ζ,φ

tin (2.113) we get for the second integral

in (2.117) the term

−λ2∫τ

φˆ

Xτ,ζ,φ

udu =−µ(τ−t) + λ2c+

βeβτ −λ2c−

βe−βτ

−λ2c+

βeβt +λ2c−

βe−βt.

(2.121)

2.4 Proofs

Plugging (2.120) and (2.121) into the sell-subgradient in (2.117) and collecting all

terms with a common exponential factor depending on tfinally yields

∇↓

tJτ(ˆ

Xτ,ζ,φ)

=eκt{−1

2ζe−2κτ −η

2e−2κτ (−c+β

κ+β+c−β

κ−β)

−η

2e−2κτ (c+β

κ+βe(κ+β)τ−c−β

κ−βe(κ−β)τ)

+η

c+β

β−κe(β−κ)τ+η

c−β

κ+βe−(κ+β)τ+1

2(ηφ ˆ

Xτ,ζ,φ

τ+ζˆ

Xτ,ζ,φ

τ)e−κτ }

+e−κt {1

2ζ−η

c+β

κ+β+η

c−β

κ−β}+eβt {η

c++β

κ+β−η

c+β

β−κ−λ2c+

β}

+e−βt {−η

c−β

κ−β−η

c−β

κ+β+λ2c−

β}

+λ2c+

βeβτ −λ2c−

βe−βτ −η

2φˆ

Xτ,ζ,φ

τ−1

2ζˆ

Xτ,ζ,φ

τ(2.122)

with φˆ

Xτ,ζ,φ

τand ζˆ

Xτ,ζ,φ

τgiven in (2.113) and (2.114), respectively. Recall that we

skipped the arguments in the functions c+(τ, ζ, φ)and c−(τ, ζ, φ). Indeed, a direct

but very tedious computation shows that all expressions in the curly brackets in

(2.122) as well as the term in the final line vanish. In other words, ∇↓

tJτ(ˆ

Xτ,ζ,φ) = 0

for all 0≤t≤τ. By virtue of Lemma 2.3.8 this implies ∇↑

tJτ(ˆ

Xτ,ζ,φ)>0for

t∈[0, τ]. As a consequence, the strategy ˆ

Xτ,ζ,φ is optimal due to the first order

optimality conditions in Proposition 2.3.7. Finally, since ˆ

Xτ,ζ,φ,↓

0=ℓ0(τ, ζ, φ)=0

as shown in 1.), we have (τ, ζ, φ)∈∂Rsell.

Proof of Corollary 2.3.18.Note that for (τ, ζ, φ)∈Ssatisfying (2.37) and x

given by (2.38), i.e.,

x=φ−µ

λ2D(τ)−κ

λζC(τ)

1 + κ

ληC(τ)>0,

we obtain the relation

φ−x=µ

λ2D(τ) + κ

λ(ζ+ηx)C(τ).

In other words, the problem data (τ, ζ +ηx, φ −x)satisfies the identity in (2.33) of

Proposition 2.3.17. Hence, we can immediately conclude from Proposition 2.3.17

and Lemma 2.3.13 that the strategy ˆ

Xτ,ζ,φ defined in (2.39) is optimal. In par-

ticular, it follows that (τ, ζ, φ)∈Rsell by the definition of the selling-region in

(2.22).

2.4.5 Proofs of Section 2.3.4

The next lemma summarizes some basic results which we will use several times in

the following section.

2 Optimal Investment with Transient Price Impact

Lemma 2.4.3. Let (τ, ζ, φ)∈S,τ≥0,ζ > 0, be some arbitrary problem data

with corresponding optimal strategy ˆ

Xτ,ζ,φ = ( ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)∈Xd. For any

θ > 0consider the problem data (τ+θ, ζeκθ, φ)∈Sand the strategy

Xτ+θ,ζeκθ,φ

t≜ˆ

Xτ,ζ,φ

t−θ1[θ,τ+θ](t) (0 ≤t≤τ+θ)(2.123)

in Xdsuch that φXτ+θ,ζeκθ,φ

0−=φXτ+θ,ζeκθ,φ

θ−=φ,ζXτ+θ,ζeκθ,φ

0−=ζeκθ and ζXτ+θ,ζeκθ,φ

θ−=

ζ.

1. Assume that

∇↑

0Jτ(ˆ

Xτ,ζ,φ) = 0.(2.124)

Then we have

w1,2(θ, ζ, φ, τ)≜∇↓,↑

0Jτ+θ(Xτ+θ,ζeκθ,φ)

=±(µ−ασ2φ)θ+1

2ζ(eκθ ±1) + 1

2η|φˆ

Xτ,ζ,φ

τ|(e−κ(τ+θ)±e−κτ )

2η(e−κθ ±1)∫[0,τ]

e−κu(dˆ

Xτ,ζ,φ,↑

u+dˆ

Xτ,ζ,φ,↓

u).

(2.125)

The maps θ↦→ w1,2(θ, ζ, φ, τ)are continuous and strictly convex.

2. Assume that

∇↓

0Jτ(ˆ

Xτ,ζ,φ) = 0.

Then we have

w3,4(θ, ζ, φ, τ)≜∇↑,↓

0Jτ+θ(Xτ+θ,ζeκθ,φ)

=±(ασ2φ−µ)θ+1

2ζ(eκθ ±1) + 1

2η|φˆ

Xτ,ζ,φ

τ|(e−κ(τ+θ)±e−κτ )

2η(e−κθ ±1)∫[0,τ]

e−κu(dˆ

Xτ,ζ,φ,↑

u+dˆ

Xτ,ζ,φ,↓

u).

(2.126)

The maps θ↦→ w3,4(θ, ζ, φ, τ)are continuous and strictly convex.

3. Assume that τ= 0 and φ= 0. Then we have

w5,6(θ, ζ, ϱ)≜ϱ∇↑,↓

0Jθ(Xθ,ζeκθ,0) = ∓µθ +1

2ζ(eκθ ±ϱ).(2.127)

The maps θ↦→ w5,6(θ, ζ, φ, τ)are continuous and strictly convex.

Proof. 1.) and 2.): We will only compute the mapping θ↦→ w1(θ, ζ, φ, τ)in (2.125).

The computations for the mappings w2, w3and w4are very similar and there-

fore omitted. Hence, let (τ, ζ, φ)∈Swith associated optimal strategy ˆ

Xτ,ζ,φ =

(ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)satisfy (2.124). We compute the sell-subgradient of the strategy

Xτ+θ,ζeκθ,φ in (2.123), i.e.,

∇↓

0Jτ+θ(Xτ+θ,ζeκθ,φ) = w1(θ, ζ, φ, τ).

2.4 Proofs

For convenience, we will henceforth write Xfor the strategy in (2.123) and denote

by φX,ζXthe corresponding stock holdings and spread dynamics. By definition

of the sell-subgradient in (2.16) we have

∇↓

0Jτ+θ(Xτ+θ,ζeκθ,φ)

=∫τ+θ

κe−κtζX

tdt +∫τ+θ

(µ−ασ2φX

t)dt +κ∫θ

e−κtζX

tdt +θ(µ−ασ2φ)

2(η|φX

τ+θ|+ζX

τ+θ)e−κ(τ+θ)−1

2ηφX

τ+θ−1

2sign(φX

τ+θ)ζX

τ+θ.(2.128)

Observe that the assumption in (2.124) implies

0 = ∇↑

0Jτ(ˆ

Xτ,ζ,φ) = ∇↑

θJτ+θ(Xτ+θ,ζeκθ,φ)

and thus gives us the identity

∫τ+θ

(µ−ασ2φX

t)dt =∫τ+θ

κe−κ(t−θ)ζX

tdt +1

2(η|φX

τ+θ|+ζX

τ+θ)e−κτ

2ηφX

τ+θ+1

2sign(φX

τ+θ)ζX

τ+θ.(2.129)

Plugging back (2.129) into (2.128) and using

κ∫θ

e−κtζX

tdt =κeκθζ∫θ

e−2κtdt =−1

2ζ(e−κθ −eκθ),

since ζX

t=ζeκ(θ−t)on [0, θ], we obtain

∇↓

0Jτ+θ(Xτ+θ,ζeκθ,φ)

=κ(1 + eκθ)∫τ+θ

ζX

te−κtdt −1

2ζ(e−κθ −eκθ) + θ(µ−ασ2φ)

2η|φX

τ+θ|(e−κτ +e−κ(τ+θ)) + 1

2ζX

τ+θ(e−κτ +e−κ(τ+θ)).(2.130)

Next, let us rewrite the integral in (2.130) by using the spread dynamics

ζX

t=ζe−κ(t−θ)+e−κ(t−θ)∫[θ,t]

ηeκ(s−θ)(dX↑

s+dX↓

s) (θ≤t≤τ+θ)(2.131)

and Fubini’s Theorem:

∫τ+θ

ζX

te−κtdt

=ζeκθ ∫τ+θ

e−2κtdt +∫[θ,τ+θ]

ηeκu (∫τ+θ

e−2κtdt)(dX↑

u+dX↓

=−1

2κζeκθ(e−2κ(τ+θ)−e−2κθ)

−η

2κ∫[θ,τ+θ]

e2κu(e−2κ(τ+θ)−e−2κu)(dX↑

u+dX↓

u).(2.132)

2 Optimal Investment with Transient Price Impact

As a consequence

κ(1 + eκθ)∫τ+θ

ζX

te−κtdt (2.133)

2e−κθ (ζ(1 −e−2κτ ) + η∫[θ,τ+θ]

eκθ(e−κu −e−2κ(τ+θ)+κu)(dX↑

u+dX↓

u))

2ζ(1 −e−2κτ ) + 1

2η∫[θ,τ+θ]

eκθ(e−κu −e−2κ(τ+θ)+κu)(dX↑

u+dX↓

u).

Moreover, using once more the representation of the spread ζX

τ+θin (2.131) we can

write

2ζX

τ+θ(e−κτ +e−κ(τ+θ))

2ζe−2κτ +1

2ηe−2κτ ∫[θ,τ+θ]

eκ(u−θ)(dX↑

u+dX↓

2ζe−2κτ−κθ +1

2ηe−2κτ−κθ ∫[θ,τ+θ]

eκ(u−θ)(dX↑

u+dX↓

u).(2.134)

Inserting (2.134) and (2.133) back into (2.130), some terms will cancel out and we

obtain

w1(θ, ζ, φ, τ) = (µ−ασ2φ)θ+1

2ζ(eκθ + 1) + 1

2η|φX

τ+θ|(e−κ(τ+θ)+e−κτ )

2η(1 + e−κθ)∫[θ,τ+θ]

eκ(θ−u)(dX↑

u+dX↓

u).

Noting that φX

τ+θ=φˆ

τand

∫[θ,τ+θ]

eκ(θ−u)(dX↑

u+dX↓

u) = ∫[0,τ]

e−κu(dˆ

X↑

u+dˆ

X↓

yields the desired result in (2.125). Obviously, the map w1is continuous in θ.

Finally, we have

∂2

∂θ2w1(θ, ζ, φ, τ) = κ2

2ζeκθ +κ2η

2|φˆ

τ|e−κ(τ+θ)

+κ2η

2e−κθ ∫[0,τ]

e−κu(dˆ

X↑

u+dˆ

X↓

u)>0 (θ≥0)

implying that θ↦→ w1(θ, ζ, φ, τ)is strictly convex because ζ > 0.

3.) Let (0, ζ, 0) ∈Swith associated optimal strategy ˆ

X0,ζ,0= (0,0); cf. Remark

2.3.16. By the definition in (2.15) and (2.16), we obtain for the buy- and sell-

subgradient of the strategy Xθ,ζeκθ,0

t= (0,0) for t∈[0, θ]in (2.123) the expressions

w5,6(θ, ζ, ϱ) = ϱ∇↑,↓

0Jθ(Xθ,ζeκθ,0)

=∓µθ +κζeκθ ∫θ

e−2κudu +1

2ζe−κθ ±1

2ϱζ

=∓µθ +1

2ζ(eκθ ±ϱ)

2.4 Proofs

as desired. In particular

∂2

∂θ2w5,6(θ, ζ, ϱ) = 1

2ζκ2eκθ >0 (θ≥0)

implying that θ↦→ w5,6(θ, ζ, ϱ)is strictly convex because ζ > 0.

A first implication of the results from Lemma 2.4.3 is presented in the next

lemma.

Lemma 2.4.4.

1. Let (τ, ζ, φ)∈∂Rbuy,τ > 0,ζ > 0, be some problem data with corresponding

optimal strategy ˆ

Xτ,ζ,φ = ( ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)∈Xdsuch that

φ≤µ

ασ2(2.135)

as well as d

dt (∇↑

tJτ(ˆ

Xτ,ζ,φ))⏐⏐⏐⏐t↓0

= 0.(2.136)

Then for any data (τ+θ, ζeκθ, φ)∈Swith θ > 0the corresponding optimal

strategy is given by

Xτ+θ,ζeκθ,φ

t=ˆ

Xτ,ζ,φ

t−θ1[θ,τ+θ](t) (0 ≤t≤τ+θ).(2.137)

In particular (τ+θ, ζeκθ, φ)∈Rwait for all θ > 0.

2. Analogously, let (τ, ζ, φ)∈∂Rsell,τ > 0,ζ > 0, be some problem data with

corresponding optimal strategy ˆ

Xτ,ζ,φ = ( ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)∈Xdsuch that

φ≥µ

ασ2(2.138)

as well as d

dt (∇↓

tJτ(ˆ

Xτ,ζ,φ))⏐⏐⏐⏐t↓0

= 0.(2.139)

Then for any data (τ+θ, ζeκθ, φ)∈Swith θ > 0the corresponding optimal

strategy is given by

Xτ+θ,ζeκθ,φ

t=ˆ

Xτ,ζ,φ

t−θ1[θ,τ+θ](t) (0 ≤t≤τ+θ).(2.140)

In particular (τ+θ, ζeκθ, φ)∈Rwait for all θ > 0.

Proof. We will only justify the assertion in 1.). Exactly the same reasoning applies

for the assertion in 2.). Hence, let (τ, ζ, φ)∈∂Rbuy,τ > 0, be some arbitrary

data with corresponding optimal strategy ˆ

Xτ,ζ,φ = ( ˆ

Xτ,ζ,φ,↑,ˆ

Xτ,ζ,φ,↓)satisfying

Xτ,ζ,φ,↑

0= 0 and ∇↑

0Jτ(ˆ

Xτ,ζ,φ)=0. Moreover, let θ > 0be arbitrary but fixed.

We consider the candidate strategy ˆ

Xτ+θ,ζeκθ,φ with data (τ+θ, ζeκθ, φ)∈Sas

2 Optimal Investment with Transient Price Impact

defined in (2.137). By definition, it holds that φˆ

Xτ+θ,ζeκθ,φ

θ−=φand ζˆ

Xτ+θ,ζeκθ,φ

θ−=ζ.

This implies that

∇↑,↓

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ) = ∇↑,↓

t−θJτ(ˆ

Xτ,ζ,φ) (θ≤t≤τ+θ)

(cf. Remark 2.3.9). Hence, in order to verify the first order optimality conditions

from Proposition 2.3.7 for the candidate strategy ˆ

Xτ+θ,ζeκθ,φ, we only have to check

the corresponding buy- and sell-subgradients on [0, θ]. By virtue of Lemma 2.4.3

1.) and the definition of ˆ

Xτ+θ,ζeκθ,φ, we have for any t∈[0, θ]the expressions

∇↓

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ) = ∇↓

0Jτ+θ−t(ˆ

Xτ+θ−t,ζeκ(θ−t),φ) = w1(θ−t, ζ, φ, τ)(2.141)

as well as

∇↑

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ) = ∇↑

0Jτ+θ−t(ˆ

Xτ+θ−t,ζeκ(θ−t),φ) = w2(θ−t, ζ, φ, τ).(2.142)

Concerning the sell-subgradient in (2.141), we observe that assumption (2.135),

i.e., φ≤µ/(ασ2), implies

∇↓

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ) = w1(θ−t, ζ, φ, τ)>0 (0 ≤t≤θ);

cf. the definition of w1in (2.125). Concerning the buy-subgradient in (2.142), first

note that we have

w2(0, ζ, φ, τ) = ∇↑

0Jτ(ˆ

Xτ,ζ,φ) = ∇↑

θJτ+θ(ˆ

Xτ+θ,ζeκθ,φ) = 0.(2.143)

Moreover, differentiability of the buy-subgradient t↦→ ∇↑

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ), cf.

the definition in (2.15), together with assumption (2.136) as well as the identity

in (2.142) implies

0 = d

dt (∇↑

tJτ(ˆ

Xτ,ζ,φ))⏐⏐⏐⏐t↓0

dt (∇↑

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ))⏐⏐⏐⏐t↑θ

dt (w2(θ−t, ζ, φ, τ))⏐⏐⏐⏐t↑θ

=−∂

∂θw2(0, ζ, φ, τ)

(2.144)

and thus ∂

∂θ w2(0, ζ, φ, τ) = 0. Consequently, since the mapping t↦→ w2(θ−t, ζ, φ, τ)

is strictly convex on [0, θ]by virtue of Lemma 2.4.3 1.), it follows from (2.143) and

(2.144) that

∇↑

tJτ+θ(ˆ

Xτ+θ,ζeκθ,φ) = w2(θ−t, ζ, φ, τ)>0 (0 ≤t < θ).

To sum up, in view of the first order optimality conditions in Proposition 2.3.7,

candidate strategy ˆ

Xτ+θ,ζeκθ,φ in (2.137) is optimal.

Proof of Proposition 2.3.20.1.): We start with the assertion in 1.). First, let us

mention that for all κ > 0the equation in (2.44) admits a unique positive solution

2.4 Proofs

θ > 0. Moreover, recall that C(τ)>0and D(τ)∈(0,1) due to Lemma 2.4.2 which

implies that ¯

ζ=s1(τ)>0. Setting ¯φ=ϕ1(τ, ¯

ζ)>0obviously yields that the

triplet (τ, ¯

ζ, ¯φ)satisfies the identity in (2.42). A direct computation reveals that

¯φ=µ

λ2−1

2κ¯

ζeκ¯

θ1−e−κ¯

λ2

as claimed in (2.46). Now, let us argue via the first order optimality conditions from

Proposition 2.3.7 that the candidate strategy Xτ+¯

θ,¯

ζeκ¯

θ,¯φin (2.43) with problem

data (τ+¯

θ, ¯

ζeκ¯

θ,¯φ), which clearly belongs to Xd, is optimal and satisfies the

properties in (2.47). By construction of Xτ+¯

θ,¯

ζeκ¯

θ,¯φ, we only have to check the

corresponding buy- and sell-subgradients on the time interval [0,¯

θ]. For this, note

that we can use the results from Lemma 2.4.3 2.) because the optimal strategy

Xτ,¯

ζ, ¯φfrom Proposition 2.3.17 satisfies ∇↓

0Jτ(ˆ

Xτ,¯

ζ, ¯φ) = 0. Hence, we obtain for the

sell-subgradient on [0,¯

θ]the expression

∇↓

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) = ∇↓

0Jτ+¯

θ−t(Xτ+¯

θ−t,¯

ζeκ(¯

θ−t),¯φ) = w4(¯

θ−t, ¯

ζ, ¯φ, τ)

with w4(0,¯

ζ, ¯φ, τ) = 0. In addition, it holds that

Xτ,¯

ζ, ¯φ,↓

0= 0 and d

dt (∇↓

tJτ(ˆ

Xτ,¯

ζ, ¯φ))⏐⏐⏐⏐t↓0

= 0 (2.145)

which implies ∂

∂θ w4(0,¯

ζ, ¯φ, τ) = 0; cf. also the argumentation in the proof of Lemma

2.4.4. Thus, appealing to the strict convexity of the mapping t↦→ w4(¯

θ−t, ¯

ζ, ¯φ, τ)

on [0,¯

θ]we can deduce that

∇↓

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) = w4(¯

θ−t, ¯

ζ, ¯φ, τ)>0 (0 ≤t < ¯

θ).(2.146)

Concerning the buy-subgradient of our candidate strategy Xτ+¯

θ,¯

ζeκ¯

θ,¯φ, we obtain

by Lemma 2.4.3 2.) the expression

∇↑

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) = ∇↑

0Jτ+¯

θ−t(Xτ+¯

θ−t,¯

ζeκ(¯

θ−t),¯φ) = w3(¯

θ−t, ¯

ζ, ¯φ, τ)

= (λ2¯φ−µ)(¯

θ−t) + 1

2¯

ζ(eκ(¯

θ−t)+ 1)

2η(e−κ(¯

θ−t)+ 1)∫[0,τ]

e−κudˆ

Xτ,¯

ζ, ¯φ,↓

u(0 ≤t≤¯

θ),

(2.147)

where we used the fact that ˆ

Xτ,¯

ζ, ¯φ,↓

τ=φˆ

Xτ,¯

ζ, ¯φ

τ, i.e., the remaining position in the

risky asset is liquidated at final time τ(recall Remark 2.3.19 2.)). Actually, the

fact that ∂

∂θ w4(0,¯

ζ, ¯φ, τ) = 0 is equivalent to the identity

∫[0,τ]

e−κudˆ

Xτ,¯

ζ, ¯φ,↓

u=2

ηκ (−λ2¯φ+µ+1

2κ¯

ζ).(2.148)

2 Optimal Investment with Transient Price Impact

Hence, rewriting the expression in (2.147) by inserting (2.148) yields

∇↑

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) = w3(¯

θ−t, ¯

ζ, ¯φ, τ)

=(λ2(¯

θ−t)−λ2

κ(e−κ(¯

θ−t)+ 1))¯φ+1

2(e−κ(¯

θ−t)+2+eκ(¯

θ−t))¯

+µ

κ(e−κ(¯

θ−t)+ 1) −µ(¯

θ−t)

(2.149)

and thus

dt∇↑

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) = −∂

∂θw3(¯

θ−t, ¯

ζ, ¯φ, τ)

=(−λ2−λ2e−κ(¯

θ−t))¯φ+κ

2(e−κ(¯

θ−t)−eκ(¯

θ−t))¯

ζ+µe−κ(¯

θ−t)+µ.

(2.150)

Now, observe that both properties in (2.47) are satisfied if and only if, respectively,

¯φ=µκ¯

θ−µ(1 + e−κ¯

θ)−1

2κ¯

ζ(e−κ¯

θ+2+eκ¯

θ)

κλ2¯

θ−λ2(1 + e−κ¯

θ)

=µ

λ2−1

2κ¯

ζeκ¯

θ(1 + e−κ¯

θ)2

λ2(κ¯

θ−(1 + e−κ¯

θ))

and

¯φ=µ(1 + e−κ¯

θ)−1

2κ¯

ζ(eκ¯

θ−e−κ¯

θ)

λ2+λ2e−κ¯

θ=µ

λ2−1

2κ¯

ζeκ¯

θ1−e−κ¯

λ2.

But this is the case due to the definition of ¯φin (2.46) as well as the fact that the

constant ¯

θsatisfies the identity in (2.44). Indeed, we have

(1 + e−κ¯

θ)2

κ¯

θ−(1 + e−κ¯

θ)= 1 −e−κ¯

if an only if eκ¯

θκ¯

θ−2−κ¯

θ−2eκ¯

θ= 0 which holds true by the definition of ¯

θ.

Moreover, referring once more to Lemma 2.4.3 2.), we know that the mapping

t↦→ w3(¯

θ−t, ¯

ζ, ¯φ, τ)is strictly convex on [0,¯

θ]. As a consequence, together with

w3(0,¯

ζ, ¯φ, τ)>0(cf. also (2.147)), w3(¯

θ, ¯

ζ, ¯φ, τ) = 0 as well as ∂

∂θ w3(¯

θ, ¯

ζ, ¯φ, τ) = 0,

we can conclude that

w3(¯

θ−t, ¯

ζ, ¯φ, τ) = ∇↑

tJτ+¯

θ(Xτ+¯

θ,¯

ζeκ¯

θ,¯φ)>0 (0 < t ≤¯

θ).(2.151)

To sum up, in view of (2.146) and (2.151), we can deduce from the first order op-

timality conditions in Proposition 2.3.7 that the strategy Xτ+¯

θ,¯

ζeκ¯

θ,¯φwith problem

data (τ+¯

θ, ¯

ζeκ¯

θ,¯φ)is optimal. By definition of the boundary of the buying-region

in (2.21), it follows that (τ+¯

θ, ¯

ζeκ¯

θ,¯φ)∈∂Rbuy.

2.): Let us now prove the assertion in 2.). Consider ζ∈[0,¯

ζ)arbitrary but fixed

and choose τζ≥0such that we have ¯φ=ϕ1(τζ, ζ), i.e., the problem data (τζ, ζ, ¯φ)

satisfies (2.42) (note that it is always possible to find such a τζin a unique manner).

2.4 Proofs

Thus, for any θ > 0, denoting by Xτζ+θ,ζeκθ,¯φthe candidate strategy in (2.43) with

problem data (τζ+θ, ζeκθ,¯φ), we obtain as above in (2.149) the expression

∇↑

0Jτζ+θ(Xτζ+θ,ζeκθ,¯φ) = w3(θ, ζ, ¯φ, τζ)

=(λ2θ−λ2

κ(e−κθ + 1))¯φ+1

2(e−κθ +2+eκθ)ζ

+µ

κ(e−κθ + 1) −µθ.

(2.152)

Observe that the map w3in (2.152) is actually independent of τζand strictly

increasing in ζ. Hence, together with the result in 1.), we obtain

0 = w3(¯

θ, ¯

ζ, ¯φ, τ)> w3(¯

θ, ζ, ¯φ, τ) = w3(¯

θ, ζ, ¯φ, τζ).

Consequently, since w3(θ, ζ, ¯φ, τζ)is continuous and strictly convex in θwith w3(0, ζ, ¯φ, τζ)>

0(recall (2.147)), there must exist a unique θ∗∈[0,¯

θ)such that

w3(θ∗, ζ, ¯φ, τζ) = ∇↑

0Jτζ+θ∗(Xτζ+θ∗,ζeκθ∗,¯φ) = 0.

Moreover, strict convexity of w3in θand the fact that ∂

∂θ w3is strictly increasing

also in ζ, cf. (2.152), we obtain together with the result in 1.) the upper estimate

0 = ∂

∂θw3(¯

θ, ¯

ζ, ¯φ, τ)>∂

∂θw3(θ∗,¯

ζ, ¯φ, τ)

>∂

∂θw3(θ∗, ζ, ¯φ, τ) = ∂

∂θw3(θ∗, ζ, ¯φ, τζ).

Put differently,

0<d

dt (∇↑

tJτζ+θ∗(Xτζ+θ∗,ζeκθ∗,¯φ))⏐⏐⏐⏐t↓0

=−∂

∂θw3(θ∗, ζ, ¯φ, τζ).(2.153)

Thus, both properties in (2.48) are satisfied by Xτζ+θ∗,ζeκθ∗,¯φ. Finally, note that

it follows by similar arguments as in 1.) that the strategy Xτζ+θ∗,ζeκθ∗,¯φ∈Xdis

optimal by virtue of the first order optimality conditions in Proposition 2.3.7, that

is,

∇↓

tJτζ+θ∗(Xτζ+θ∗,ζeκθ∗,¯φ) = w4(θ∗−t, ζ, ¯φ, τζ)>0 (0 ≤t < θ∗)

and

∇↑

tJτζ+θ∗(Xτζ+θ∗,ζeκθ∗,¯φ) = w3(θ∗−t, ζ, ¯φ, τζ)>0 (0 < t ≤θ∗).

Thus, by definition of the boundary of the buying-region in (2.21), we have that

(τζ+θ∗, ζeκθ∗,¯φ)∈∂Rbuy.

3.): It remains to prove the assertion in 3.). Observe that for any ζ > ¯

ζwith

corresponding τζsuch that ¯φ=ϕ1(τζ, ζ), i.e., (τζ, ζ, ¯φ)satisfies (2.42), we obtain

once more due to strict monotonicity of w3in (2.152) in ζand the result in 1.) the

lower bound

∇↑

0Jτζ+θ(Xτζ+θ,ζeκθ,¯φ) = w3(θ, ζ, ¯φ, τζ)> w3(θ, ¯

ζ, ¯φ, τ)≥0

2 Optimal Investment with Transient Price Impact

for all θ > 0. Thus, property (2.49) is satisfied. Again, it follows by the same

arguments as in 1.) and 2.) that the strategy Xτζ+θ,ζeκθ,¯φ∈Xdis optimal for any

θ > 0by virtue of the first order optimality conditions from Proposition 2.3.7. In

particular, it holds that (τζ+θ, ζeκθ,¯φ)∈Rwait by the definition of the waiting

region in (2.24).

Lemma 2.4.5. The map τ↦→ s1(τ)in (2.45)is strictly decreasing in τ≥0.

Proof. Differentiating s1with respect to τyields s′

1(τ) = f(τ)/g(τ)with

f(τ) = −4√αe

√ακστ

√ηκ+ασ2ηκµσ{(e

2√ακστ

√ηκ+ασ2−1)(eκ¯

θ−1)ηκ

+ 2√αeκ¯

θσ(√ασ (e

2√ακστ

√ηκ+ασ2−1)

+(1 + e

2√ακστ

√ηκ+ασ2)√ηκ +ασ2)}<0.

and

g(τ) = √ηκ +ασ2{(1 + e

2√ακστ

√ηκ+ασ2)(eκ¯

θ−1)ηκ + 2√αeκ¯

θσ

(√ασ (1 + e

2√ακστ

√ηκ+ασ2)+(e

2√ακστ

√ηκ+ασ2−1)√ηκ +ασ2)}2

as desired.

Proof of Corollary 2.3.21.Let ϑ > 0and let ˆ

Xτ+¯

θ+ϑ,ζ′,φ′denote the strategy

with problem data (τ+¯

θ+ϑ, ζ′, φ′), which is defined on the interval [0, ϑ]as in (2.53)

with ζ′,φ′as defined in (2.54). By definition it holds that φˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−=φ′+

Xτ+¯

θ+ϑ,ζ′,φ′,↑

ϑ−= ¯φand ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−=¯

ζeκ¯

θ, that is, the identities in (2.52) hold true.

In order to prove Corollary 2.3.21, we proceed in two steps. First, we argue that

Xτ+¯

θ+ϑ,ζ′,φ′satisfies the first order optimality conditions from Proposition 2.3.7.

Then, we show that the strategy ˆ

Xτ+¯

θ+ϑ,ζ′,φ′belongs to Xdand that ζ′≥¯

ζeκ¯

θ>0

as claimed in (2.54).

Step 1. In view of the dynamic programming principle form Remark 2.3.9, the

strategy ˆ

Xτ+¯

θ+ϑ,ζ′,φ′coincides with the strategy ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φon [ϑ, τ +¯

θ+ϑ]in the

sense that

Xτ+¯

θ+ϑ,ζ′,φ′

t=ˆ

Xτ+¯

θ+ϑ,ζ′,φ′

ϑ−+ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φ

t−ϑ(ϑ≤t≤τ+¯

θ+ϑ).(2.154)

In particular, it holds that

∇↑,↓

tJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′) = ∇↑,↓

t−ϑJτ+¯

θ(ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) (ϑ≤t≤τ+θ+ϑ).

2.4 Proofs

As a consequence, we only have to check the corresponding buy- and sell-subgradients

of ˆ

Xτ+¯

θ+ϑ,ζ′,φ′on [0, ϑ]. To alleviate the notation, we will henceforth write φˆ

X,ζˆ

X↑and ˆ

X↓instead of φˆ

Xτ+¯

θ+ϑ,ζ′,φ′,ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′,ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑and ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↓,

respectively. By virtue of Lemma 2.3.8, the verification of the optimality of the

strategy ˆ

Xτ+¯

θ+ϑ,ζ′,φ′on [0, ϑ]reduces to show that the buy-subgradient vanishes

on [0, ϑ]. Following the definition in (2.15), the buy-subgradient is given by

∇↑

tJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′)(2.155)

=∫ϑ

t(κe−κ(u−t)ζˆ

u+λ2φˆ

u−µ)du +∫τ+¯

θ+ϑ

ϑ(κe−κ(u−t)ζˆ

u+λ2φˆ

u−µ)du

2(η|φˆ

τ+¯

θ+ϑ|+ζˆ

τ+¯

θ+ϑ)e−κ(τ+¯

θ+ϑ−t)+η

2φˆ

τ+¯

θ+ϑ+1

2sign(φˆ

τ+¯

θ+ϑ)ζˆ

τ+¯

θ+ϑ

for all 0≤t≤ϑ. Let us rewrite the buy-subgardient in a more convenient form.

Due to the first property of ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φin (2.47), we have

0 = ∇↑

0Jτ+¯

θ(ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φ) = ∇↑

ϑJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′).

This gives us the identity

∫τ+¯

θ+ϑ

(λ2φˆ

u−µ)du

=−∫τ+¯

θ+ϑ

κe−κ(u−ϑ)ζˆ

udu −1

2(η|φˆ

τ+¯

θ+ϑ|+ζˆ

τ+¯

θ+ϑ)e−κ(τ+¯

θ)

−1

2ηφ ˆ

τ+¯

θ+ϑ−1

2sign(φˆ

τ+¯

θ+ϑ)ζˆ

τ+¯

θ+ϑ.

(2.156)

Inserting (2.156) back into the buy-subgradient in (2.155) yields

∇↑

tJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′) =∫ϑ

t(κe−κ(u−t)ζˆ

u+λ2φˆ

u−µ)du

+∫τ+¯

θ+ϑ

κ(e−κ(u−t)−e−κ(u−ϑ))ζˆ

udu

2η|φˆ

τ+¯

θ+ϑ|(e−κ(τ+¯

θ+ϑ−t)−e−κ(τ+¯

θ))

2ζˆ

τ+¯

θ+ϑ(e−κ(τ+¯

θ+ϑ−t)−e−κ(τ+¯

θ)).

(2.157)

Next, let us rewrite the first integral in (2.157) by using the spread dynamics

ζˆ

u=¯

ζeκ(¯

θ+ϑ−u)−ηe−κu ∫ϑ

eκsdˆ

X↑

s(0 ≤u≤ϑ).

2 Optimal Investment with Transient Price Impact

An application of Fubini’s Theorem then yields

∫ϑ

t(κe−κ(u−t)ζˆ

u+λ2φˆ

u−µ)du

=λ2∫ϑ

φˆ

udu −µ(ϑ−t) + κeκt ∫ϑ

e−κuζˆ

udu

=λ2∫ϑ

φˆ

udu −µ(ϑ−t) + 1

2¯

ζeκ(¯

θ+ϑ+t)(e−2κt −e−2κϑ)

2η∫ϑ

(e−κ(u−t)−e−κ(t−u))dˆ

X↑

(2.158)

Moreover, let us rewrite the second integral in (2.157) by using the spread dynamics

ζˆ

u=¯

ζeκ(¯

θ+ϑ−u)on [ϑ, ¯

θ+ϑ]. We get

∫τ+¯

θ+ϑ

κ(e−κ(u−t)−e−κ(u−ϑ))ζˆ

udu

=∫¯

θ+ϑ

κ(e−κ(u−t)−e−κ(u−ϑ))ζˆ

udu +∫τ+¯

θ+ϑ

κ(e−κ(u−t)−e−κ(u−ϑ))ζˆ

udu

2¯

ζ(eκ¯

θ−e−κ¯

θ)(eκ(t−ϑ)−1) + κ(eκt −eκϑ)∫τ+¯

θ+ϑ

ϑ+¯

e−κuζˆ

udu.

For the last integral, we can use the computations from equation (2.132) above and

obtain

∫τ+¯

θ+ϑ

κ(e−κ(u−t)−e−κ(u−ϑ))ζˆ

udu

2¯

ζ(eκ¯

θ−e−κ¯

θ)(eκ(t−ϑ)−1) (2.159)

2eκt(¯

ζ(e−κ(¯

θ+ϑ)−e−κ(¯

θ+ϑ)−2κτ))

+η∫[¯

θ+ϑ,τ+¯

θ+ϑ]

(e−κu −e−2κ(τ+¯

θ+ϑ)+κu)dˆ

X↓

−1

2¯

ζeκϑ(e−κ(¯

θ+ϑ)−e−κ(¯

θ+ϑ)−2κτ))

−1

2ηeκϑ ∫[¯

θ+ϑ,τ+¯

θ+ϑ]

(e−κu −e−2κ(τ+¯

θ+ϑ)+κu)dˆ

X↓

Finally, let us rewrite the spread ζˆ

τ+¯

θ+ϑin the last line in (2.157) by using the

representation

ζˆ

τ+¯

θ+ϑ=¯

ζe−κτ +e−κτ ∫[¯

θ+ϑ,τ+¯

θ+ϑ]

ηeκ(u−(ϑ+¯

θ))dˆ

X↓

=¯

ζe−κτ +e−κτ ∫[0,τ]

ηeκudˆ

Xτ,¯

ζ, ¯φ,↓

u,(2.160)

where we used the fact that ˆ

X↓=ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↓coincides on the interval [¯

θ+

ϑ, τ +¯

θ+ϑ]with the optimal sell-only strategy ˆ

Xτ,¯

ζ, ¯φ,↓from Proposition 2.3.17

2.4 Proofs

with problem data (τ, ¯

ζ, ¯φ); cf., equation (2.154) above and the definition of the

optimal strategy ˆ

Xτ+¯

θ,¯

ζ, ¯φfrom Proposition 2.3.20 1.). Inserting (2.160) in the last

line of (2.157) yields

2ζˆ

τ+¯

θ+ϑ(e−κ(τ+¯

θ+ϑ−t)−e−κ(τ+¯

θ))

2¯

ζe−2κ(τ+¯

θ+ϑ)eκteκ(¯

θ+ϑ)+1

2ηe−2κ(τ+¯

θ+ϑ)eκt ∫[0,τ]

eκ(u+¯

θ+ϑ)dˆ

Xτ,¯

ζ, ¯φ,↓

−1

2¯

ζe−2κ(τ+¯

θ+ϑ)eκϑeκ(¯

θ+ϑ)−1

2ηe−2κ(τ+¯

θ)∫[0,τ]

eκ(u+¯

θ)dˆ

Xτ,¯

ζ, ¯φ,↓

u.(2.161)

As a result, plugging (2.158), (2.159) and (2.161) into (2.157), we obtain for the

buy-subgradient on the interval [0, ϑ]the expression

∇↑

tJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′)

=λ2∫ϑ

φˆ

udu +1

2η∫ϑ

(e−κ(u−t)−eκ(u−t))dˆ

X↑

u−µ(ϑ−t)

2¯

ζ(eκ(¯

θ+ϑ−t)−eκ¯

θ) + 1

2η(eκt −eκϑ)∫[0,τ]

e−κ(u+¯

θ+ϑ)dˆ

Xτ,¯

ζ, ¯φ,↓

2η|φˆ

τ+¯

θ+ϑ|(e−κ(τ+¯

θ+ϑ−t)−e−κ(τ+¯

θ)).

(2.162)

Now, it is left to show that the buy-subgradient in (2.162) vanishes on [0, ϑ]. There-

fore, let us use the explicit definition of the strategy ˆ

Xτ+¯

θ+ϑ,ζ′,φ′in (2.53) and com-

pute the first two integrals in (2.162) above. To alleviate the notation, we will omit

the arguments in the function c, i.e., we let c=c(¯

ζeκ¯

θ,¯φ). For the first integral,

we get

λ2∫ϑ

φˆ

udu

=λ2eβt (−1

2(¯φ−µ

λ2+c

β)e−βϑ

β)+λ2e−βt (1

2(¯φ−µ

λ2−c

β)eβϑ

β)

+µ(ϑ−t) + λ2c

β2.

For the second integral, we obtain

2η∫ϑ

(e−κ(u−t)−eκ(u−t))dˆ

X↑

=η

2eκt (β

2(¯φ−µ

λ2+c

β)e−κϑ

β−κ+β

2(¯φ−µ

λ2−c

β)e−κϑ

β+κ)

+η

2e−κt (−β

2(¯φ−µ

λ2+c

β)eκϑ

κ+β+β

2(¯φ−µ

λ2−c

β)eκϑ

κ−β)

+η

2eβt (−β

2(¯φ−µ

λ2+c

β)e−βϑ

β−κ+β

2(¯φ−µ

λ2+c

β)e−βϑ

β+κ)

+η

2e−βt (−β

2(¯φ−µ

λ2−c

β)eβϑ

κ+β−β

2(¯φ−µ

λ2−c

β)eβϑ

κ−β).

2 Optimal Investment with Transient Price Impact

Substituting both integrals in the buy-subgradient in (2.162) by the above com-

puted expressions and collecting all terms with common exponential factor depend-

ing on t, we obtain for all 0≤t≤ϑthe representation

∇↑

tJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′)

=η

2eκt{β

2(¯φ−µ

λ2+c

β)e−κϑ

β−κ+β

2(¯φ−µ

λ2−c

β)e−κϑ

β+κ

+∫[0,τ]

e−κ(u+¯

θ+ϑ)dˆ

Xτ,¯

ζ, ¯φ,↓

u+|φˆ

τ+¯

θ+ϑ|e−κ(τ+¯

θ+ϑ)}

+η

2e−κt{−β

2(¯φ−µ

λ2+c

β)eκϑ

κ+β+β

2(¯φ−µ

λ2−c

β)eκϑ

κ−β

η¯

ζeκ(¯

θ+ϑ)}

+eβt

2{−ηβ

2(¯φ−µ

λ2+c

β)e−βϑ

β−κ+ηβ

2(¯φ−µ

λ2+c

β)e−βϑ

β+κ

−λ2(¯φ−µ

λ2+c

β)e−βϑ

β}

+e−βt

2{−ηβ

2(¯φ−µ

λ2−c

β)eβϑ

β+κ−ηβ

2(¯φ−µ

λ2−c

β)eβϑ

κ−β

+λ2(¯φ−µ

λ2−c

β)eβϑ

β}

+λ2c

β2−1

2¯

ζeκ¯

θ−η

2∫[0,τ]

e−κ(u+¯

θ)dˆ

Xτ,¯

ζ, ¯φ,↓

u−η

2|φˆ

τ+¯

θ+ϑ|e−κ(τ+¯

θ).

(2.163)

The last term in (2.163) vanishes if we have

c=β2

2λ2(¯

ζeκ¯

θ+η|φˆ

τ+¯

θ+ϑ|e−κ(τ+¯

θ)+η∫[0,τ]

e−κ(u+¯

θ)dˆ

Xτ,¯

ζ, ¯φ,↓

u).(2.164)

Indeed, using the second property in (2.47) of the optimal strategy ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φ

as well as the expression of the buy-subgradient of ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φderived in (2.147)

above, we have that

0 = d

dt (∇↑

tJτ+¯

θ(ˆ

Xτ+¯

θ,¯

ζeκ¯

θ,¯φ))⏐⏐⏐⏐t↓0

=−λ2¯φ+µ−1

2κ¯

ζeκ¯

θ+1

2κηe−κ(τ+¯

θ)|φˆ

τ+¯

θ+ϑ|

2κηe−κ¯

θ∫[0,τ]

e−κudˆ

Xτ,¯

ζ, ¯φ,↓

and thus the identity

∫[0,τ]

e−κ(u+¯

θ)dˆ

Xτ,¯

ζ, ¯φ,↓

κη (λ2¯φ−µ+1

2κ¯

ζeκ¯

θ−1

2κηe−κ(τ+¯

θ)|φˆ

τ+¯

θ+ϑ|).

(2.165)

2.4 Proofs

Inserting (2.165) into (2.164) yields

c=β2

λ2(¯

ζeκ¯

θ+1

κ(λ2¯φ−µ))=c(¯

ζeκ¯

θ,¯φ)

as desired. Concerning the terms in (2.163) with common factor eβt and e−βt, a

direct computation shows that these terms actually vanish. We omit this simple

algebraic manipulation. In addition, exploiting further the identity in (2.165), one

can also easily verify that the terms in (2.163) with common factor eκt and e−κt

vanish. Again, these simple but tedious algebraic manipulations are left out. In

summary, we obtain ∇↑

tJτ+¯

θ+ϑ(ˆ

Xτ+¯

θ+ϑ,ζ′,φ′) = 0 for all 0≤t≤ϑas desired.

Step 2. To finalize the proof of Corollary 2.3.21, it is left to argue that the

strategy ˆ

Xτ+¯

θ+ϑ,ζ′,φ′belongs to Xdand that ζ′≥¯

ζeκ¯

θas claimed in (2.54). In

order to show this, let us start with two observations. First, thanks to the identity

in (2.164), it holds that c(¯

ζeκ¯

θ,¯φ)>0. Consequently, due to the fact that ¯φis

given by (2.46) which implies ¯φ < µ/λ2, we immediately obtain

¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β<0.(2.166)

Moreover, exploiting further the relation of the pair (¯

ζeκ¯

θ,¯φ)given in (2.45) and

(2.46), it can also be shown that

¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β<0(2.167)

holds true. As a consequence, we obtain

2(¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)e−β(ϑ−t)≥β

2(¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)

>β

2(¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)≥(¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)eβ(ϑ−t),

which implies

dt ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑

t=β

2(¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)e−β(ϑ−t)

−β

2(¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)eβ(ϑ−t)>0 (0 ≤t≤ϑ),

i.e., ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑is non-decreasing. In addition, since ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑

0= 0, we

have that ˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑

t≥0on [0, ϑ]. Therfore, it holds that ˆ

Xτ+¯

θ+ϑ,ζ′,φ′∈Xd.

2 Optimal Investment with Transient Price Impact

Concerning the spread ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′, we obtain on [0, ϑ]the expression

ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

t=¯

ζeκ(¯

θ+ϑ−t)−ηe−κt ∫ϑ

eκudˆ

Xτ+¯

θ+ϑ,ζ′,φ′,↑

=ηβ

2eβt (¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)e−βϑ

κ+β

−ηβ

2e−βt (¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)eβϑ

κ−β,

where we used the fact that

0 = −β

2(¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)eκϑ

κ+β

+β

2(¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)eκϑ

κ−β+¯

ζeκ(¯

θ+ϑ),

as it has been employed already for the term in (2.163) above with common factor

e−κt. Therefore, it holds that

dtζˆ

Xτ+¯

θ+ϑ,ζ′,φ′

t=ηβ2

2eβt (¯φ−µ

λ2+c(¯

ζeκ¯

θ,¯φ)

β)e−βϑ

κ+β

+ηβ2

2e−βt (¯φ−µ

λ2−c(¯

ζeκ¯

θ,¯φ)

β)eβϑ

κ−β<0 (0 ≤t≤ϑ),

due to (2.166) and (2.167). In other words, the spread ζˆ

Xτ+¯

θ+ϑ,ζ′,φ′is strictly

decreasing on [0, ϑ]. This implies in particular that ζ′≥¯

ζeκ¯

θas claimed in (2.54).

Finally, it is straightforward to verify that the corresponding optimal share holdings

φˆ

Xτ+¯

θ+ϑ,ζ′,φ′satisfy the second order ODE in (2.29) on (0, ϑ). The assertion in

(2.55) follows from Lemma 2.3.14.

Before we continue with the proof of Proposition 2.3.23, let us collect some useful

properties of the maps s2and ϕ2defined in (2.60) and (2.61), respectively.

Lemma 2.4.6.

1. On the interval [θ,¯

θ], the map θ↦→ s2(θ)in (2.60)is strictly decreasing and

the map θ↦→ ϕ2(θ)in (2.61)is stricly increasing.

2. The pair (s2(¯

θ), ϕ2(¯

θ)) coincides with the pair (s1(0), ϕ1(0, s1(0))) from Propo-

sition 2.3.20 1.) for τ= 0. In particular, (s2(¯

θ), ϕ2(¯

θ)) satisfies the relation

ϕ2(¯

θ) = 2µ+κs2(¯

θ)

2λ2+κη .

3. We have (s2(θ), ϕ2(θ)) = (e−κθ2µ/κ, 0).

2.4 Proofs

Proof. 1.) Differentiating s2with respect to θyields

s′

2(θ) = −2κ2ηµ(η+ηeκθ + 2λ2θeκθ)(e2κθ + 2κθeκθ −1)

(κη(1 + eκθ)2+ 2λ2eκθ(eκθ(κθ −1) −1))2<0.

Differentiating ϕ2with respect to θyields

ϕ′

2(θ) = 2κ2ηµeκθ(1 + eκθ)(e2κθ + 2κθeκθ −1)

(κη(1 + eκθ)2+ 2λ2eκθ(eκθ(κθ −1) −1))2>0.

2.) Let τ= 0 in Proposition 2.3.20 1.) and let (s1(0), ϕ1(0, s1(0))) denote the

corresponding pair. Then, we obtain in (2.45) together with Lemma 2.4.2

s1(0) = µ(1 −D(0))

λκC(0) + 1

2κeκ¯

θ(1 −e−κ¯

θ)=µη

λ2+1

2eκ¯

θ(1 −e−κ¯

θ)(2λ2+κη).(2.168)

Moreover, inserting (2.168) into (2.46) yields

ϕ1(0, s1(0)) = eκ¯

λeκ¯

θ+1

2κηeκ¯

θ−1

2κη.(2.169)

Since ¯

θsatisfies (2.44), a simple but tedious computation reveals that s1(0) = s2(¯

θ)

and ϕ1(0, s1(0)) = ϕ2(¯

θ). Finally, by definition we have

ϕ2(¯

θ) = ϕ1(0, s1(0)) = µ

λ2D(0) + κ

λs1(0)C(0) = 2µ+κs1(0)

2λ2+κη =2µ+κs2(¯

θ)

2λ2+κη

as claimed.

3.) Since θsatisfies (2.59), we immediately get ϕ2(θ) = 0. Moreover, it fol-

lows from the computations in the proof of Proposition 2.3.23 below that the pair

(s2(θ), ϕ2(θ)) satisfies for any θ∈[θ, ¯

θ]the relation

s2(θ) = e−κθ 2ϕ2(θ)

κ(1

2κηe−κθ −λ2)+e−κθ 2µ

κ,

see equation (2.173) below. Hence, s2(θ) = 2µe−κθ/κ.

Proof of Proposition 2.3.23.1.) We start with the assertion in 1.). First, note

that for all κ > 0the equation in (2.59) admits a unique solution θ∈(0,¯

θ). Now,

let θ∈[θ, ¯

θ]be arbitrary but fixed and let ¯

ζ=s2(θ)and ¯φ=ϕ2(θ). It follows from

Lemma 2.4.6 1.) and 2.) above that ¯

ζ > 0and ¯φ≥0satisfy

¯φ < ϕ2(¯

θ) = 2µ+κs2(¯

θ)

2λ2+κη <2µ+κ¯

2λ2+κη.

In other words, the triplet (0,¯

ζ, ¯φ)∈Ssatisfies the desired relation in (2.56).

Next, let us argue via the first order optimality conditions from Proposition 2.3.7

that the candidate strategy Xθ,¯

ζeκθ,¯φin (2.57), which clearly belongs to Xd, is

optimal and satisfies the desired properties in (2.62). Our arguments follow the

2 Optimal Investment with Transient Price Impact

same line as in the proof of Proposition 2.3.20 1.) above. Due to Lemma 2.4.3 2.),

which is also applicable in the current setting with τ= 0 (note that the optimal

strategy ˆ

X0,¯

ζ, ¯φwith data (0,¯

ζ, ¯φ)satisfies 1∇0J0(ˆ

X0,¯

ζ, ¯φ) = 0; cf. Remark 2.3.16),

we obtain for the sell-subgradient on [0, θ]the expression

1∇↓

tJθ(Xθ,¯

ζeκθ,¯φ) = 1∇↓

0Jθ−t(Xθ−t,¯

ζeκ(θ−t),¯φ) = w4(θ−t, ¯

ζ, ¯φ, 0)

=−(λ2¯φ−µ)(θ−t) + 1

2¯

ζ(eκ(θ−t)−1) + 1

2η¯φ(e−κ(θ−t)−1) (2.170)

with w4(0,¯

ζ, ¯φ, 0) = 0. In addition, since (0,¯

ζ, ¯φ)satisfies the relation in (2.56),

we get the upper bound

1∇↓

tJθ(Xθ,¯

ζeκθ,¯φ)≤ −µ−1

2κ¯

ζ+ ¯φ(λ2+1

2ηκ)<0,

which implies 1∇↓

tJθ(Xθ,¯

ζeκθ,¯φ)>0on [0, θ). Concerning the buy-subgradient on

[0, θ]of our candidate strategy Xθ,¯

ζeκθ,¯φin (2.57), we obtain by Lemma 2.4.3 2.)

the expression

1∇↑

tJθ(Xθ,¯

ζeκθ,¯φ) = 1∇↑

0Jθ−t(Xθ−t,¯

ζeκ(θ−t),¯φ) = w3(θ−t, ¯

ζ, ¯φ, 0)

= (λ2¯φ−µ)(θ−t) + 1

2¯

ζ(eκ(θ−t)+ 1) + 1

2η¯φ(e−κ(θ−t)+ 1) (2.171)

as well as

1∇↑

tJθ(Xθ,¯

ζeκ¯

θ,¯φ) = −∂

∂θw3(θ−t, ¯

ζ, ¯φ, 0)

=−λ2¯φ+µ−1

2κ¯

ζeκ(θ−t)+1

2κη ¯φe−κ(θ−t).

(2.172)

Now, observe that both properties in (2.62) are satisfied if, and only if

¯φ=µθ −1

2¯

ζeκθ (1 + e−κθ)

λ2θ+1

2η(1 + e−κθ)and ¯

ζeκθ =2¯φ

κ(1

2κηe−κθ −λ2)+2µ

κ(2.173)

which is in fact equivalent to ¯φ=ϕ2(θ)and ¯

ζ=s2(θ). Moreover, referring once

more to Lemma 2.4.3 2.), we know that the mapping t↦→ w3(θ−t, ¯

ζ, ¯φ, 0) is strictly

convex on [0, θ]. As a consequence, together with w3(0,¯

ζ, ¯φ, 0) >0,w3(θ, ¯

ζ, ¯φ, 0) =

0as well as ∂

∂θ w3(θ, ¯

ζ, ¯φ, 0) = 0, we can conclude that

1∇↑

tJθ(Xθ,¯

ζeκθ,¯φ) = w3(θ−t, ¯

ζ, ¯φ, 0) >0 (0 < t ≤θ).

In other words, in view of the first order optimality conditions from Proposi-

tion 2.3.7, we obtain that the strategy Xθ,¯

ζeκθ,¯φwith problem data (θ, ¯

ζeκθ,¯φ)

is optimal. In particular, we have that (θ, ¯

ζeκθ,¯φ)∈∂Rbuy.

2.): Let us now prove the assertion in 2.). Consider ζ∈[max{0,(¯φ(2λ2+κη)−

2µ)/κ},¯

ζ]. Then, it follows that (0, ζ, ¯φ)satisfies the relation in (2.56). Moreover,

2.4 Proofs

denoting by Xθ,ζeκθ,¯φthe candidate strategy in (2.57) with data (θ, ζeκθ,¯φ), we

obtain as above in (2.171) for the buy-subgradient the representation

1∇↑

0Jθ(Xθ,ζeκθ,¯φ) = w3(θ, ζ, ¯φ, 0)

= (λ2¯φ−µ)θ+1

2ζ(eκθ + 1) + 1

2η¯φ(e−κθ + 1).(2.174)

Note that the map ζ↦→ w3(θ, ζ, ¯φ, 0) above as well as ζ↦→ ∂

∂θ w3(θ, ζ, ¯φ, 0) are

strictly increasing in ζ. Thus, it follows exactly by the same arguments as in the

proof of Proposition 2.3.20 2.) that there exists a unique θ∗∈[0, θ)such that the

candidate strategy Xθ∗,ζeκθ∗,¯φin (2.57) with problem data (θ∗, ζeκθ∗,¯φ)is optimal

and satisfies the desired properties in (2.63), i.e., w3(θ∗, ζ, ¯φ, 0) = 0.

3.): It remains to prove the assertion in 3.). Obviously, for any ζ > ¯

ζthe triplet

(0, ζ, ¯φ)satisfies the desired relation in (2.56). In fact, the rest follows exactly as

in the proof of Proposition 2.3.20 3.).

Proof of Corollary 2.3.24.The proof is very similar to the proof of Corol-

lary 2.3.21 above. Therefore, we only summarize the main steps. Let ϑ > 0and let

Xθ+ϑ,ζ′,φ′denote the strategy from Corollary 2.3.24 with associated problem data

(θ+ϑ, ζ′, φ′).

First, as in the proof of Corollary 2.3.21, step 1, we have to show that ˆ

Xθ+ϑ,ζ′,φ′

satisfies the first order optimality conditions from Proposition 2.3.7. Due to the

dynamic programming principle form Remark 2.3.9, we only have to check the

corresponding buy- and sell-subgradients on [0, ϑ]. In addition, by virtue of Lemma

2.3.8, we merely have to convince ourselves that the buy-subgradient of ˆ

Xθ+ϑ,ζ′,φ′

vanishes on [0, ϑ]. As in the proof of Corollary 2.3.21, we write φˆ

X,ζˆ

X,ˆ

X↑and

X↓instead of φˆ

Xθ+ϑ,ζ′,φ′,ζˆ

Xθ+ϑ,ζ′,φ′,ˆ

Xθ+ϑ,ζ′,φ′,↑and ˆ

Xθ+ϑ,ζ′,φ′,↓, respectively, in

order to alleviate the notation. Observe that by the same computations as in step

1 in the proof of Corollary 2.3.21, it holds that

∇↑

tJθ+ϑ(ˆ

Xθ+ϑ,ζ′,φ′)

=λ2∫ϑ

φˆ

udu +1

2η∫ϑ

(e−κ(u−t)−eκ(u−t))dˆ

X↑

u−µ(ϑ−t)

2¯

ζ(eκ(θ+ϑ−t)−eκθ) + 1

2η¯φ(e−κ(θ+ϑ−t)−e−κθ)(0 ≤t≤ϑ);

(2.175)

cf. the representation of the buy-subgradient in (2.162) above with τ= 0. Comput-

ing the first two integrals in (2.175) by using the explicit representation of ˆ

Xθ+ϑ,ζ′,φ′

on [0, ϑ]as given in Corollary 2.3.21, definition (2.53), yields exactly the same rep-

resentation of the buy-subgradient as in (2.163) above with τ= 0 and function c

as defined in (2.51) evaluated at (¯

ζeκθ,¯φ). Specifically, note that c(¯

ζeκθ,¯φ)must

satisfy

c(¯

ζeκθ,¯φ) = β2

2λ2(¯

ζeκθ +η¯φe−κθ);(2.176)

2 Optimal Investment with Transient Price Impact

cf. also the equation in (2.164) above with τ= 0. Indeed, analogously to the proof

of Corollary 2.3.21, the second property in (2.62) of the optimal strategy ˆ

Xθ,¯

ζeκθ,¯φ

from Proposition 2.3.23 1.) implies the identity

η¯φe−κθ =2

κ(λ2¯φ−µ+1

2κ¯

ζeκθ);(2.177)

cf. the identity in (2.165) with τ= 0. Thus, inserting (2.177) into (2.176) yields

c(¯

ζeκθ,¯φ) = β2

λ2(¯

ζeκθ +1

κ(λ2¯φ−µ)),

as desired. In addition, as in step 1 in the proof of Corollary 2.3.21 above, one can

deduce after some simple algebraic manipulations that ∇↑

tJθ+ϑ(ˆ

Xθ+ϑ,ζ′,φ′) = 0 in

fact holds true for all 0≤t≤ϑ.

Finally, following the same line of computations as in the proof of Corollary

2.3.21 step 2, it can be shown that ˆ

Xθ+ϑ,ζ′,φ′,↑is non-negative and increasing and

that ζ′≥¯

ζeκθ.

Lemma 2.4.7. Let ϱ∈[−1,1] and let θϱ≥0denote the solution to the equation in

(2.68). Then, it holds that θϱis unique and strictly increasing in ϱ. In particular,

we have θϱ∈[θ−1, θ1] = [0, θ]with θ > 0from equation (2.59).

Proof. The solution θϱto the equation in (2.68) is given by

θϱ=1 + w(ϱe−1)

κ,

where w(z)is the principal solution for the equation z=w(z)ew(z). It is well

known that wis strictly increasing on [−e−1, e]. Moreover, it holds that θ−1= 0

and θ1=θfrom equation (2.59).

Proof of Proposition 2.3.25.1.) We start with the assertion in 1.). First, due to

Lemma 2.4.7, note that for all κ > 0the equation in (2.68) admits a unique solution

θϱ. Moreover, it holds that θϱ∈[θ−1, θ1] = [0, θ]. Now, let ϱ∈[−1,1] be arbitrary

but fixed and let ¯

ζϱ=s3(ϱ) = e−κθϱ2µ/κ as in (2.69). We have to show via the

first order optimality conditions from Proposition 2.3.7 that the candidate strategy

Xθϱ,¯

ζϱeκθϱ,0in (2.66) with problem data (θϱ,¯

ζϱeκθϱ,0) = (θϱ,2µ/κ, 0) is optimal.

By Lemma 2.4.3 3.) we have for the sell-subgradient on [0, θϱ]the representation

ϱ∇↓

tJθϱ(Xθϱ,¯

ζϱeκθϱ,0) = ϱ∇↓

0Jθϱ(Xθϱ−t,¯

ζϱeκ(θϱ−t),0) = w6(θϱ−t, ¯

ζϱ, ϱ)

=µ(θϱ−t) + 1

2¯

ζϱ(eκ(θϱ−t)−ϱ)≥0.(2.178)

Moreover, the buy-subgradient on [0, θϱ]is given by

ϱ∇↑

tJθϱ(Xθϱ,¯

ζϱeκθϱ,0) = w5(θϱ−t, ¯

ζϱ, ϱ)

=−µ(θϱ−t) + 1

2¯

ζϱ(eκ(θϱ−t)+ϱ)

2.4 Proofs

with derivative

ϱ∇↑

tJθϱ(Xθϱ,¯

ζϱeκθϱ,0) = −∂

∂θw5(θϱ−t, ¯

ζϱ, ϱ) = µθϱ−1

2κ¯

ζϱeκ(θϱ−t).

Now, observe that both properties in (2.70) are satisfied if, and only if

ζϱeκθϱ=2µθϱ

ϱe−κθϱ+ 1 and ¯

ζϱeκθϱ=2µ

κ,

which is the case because θϱsatisfies the equation in (2.68). Consequently, since

the mapping t↦→ w5(θϱ−t, ¯

ζϱ, ϱ)is strictly convex on [0, θϱ], we can deduce that

ϱ∇↑

tJθϱ(Xθϱ,2µ/κ,0)>0on (0, θϱ]. Thus, together with (2.178), it follows from the

first order optimality conditions in Proposition 2.3.7 that Xθϱ,2µ/κ,0is optimal.

2.) Note that θ1=θclearly satisfies the equation in (2.59) in Proposition 2.3.23.

In addition, we have ¯

ζ1=s3(1) = e−κθ2µ/κ =s2(θ)as well as ϕ2(θ) = 0 due to

Lemma 2.4.6 3.). As a consequence, for any ζ∈[0,¯

ζ1) = [0, s2(θ)), we are back in

the situation of Proposition 2.3.23 2.) with θ=θas claimed.

3.) Let ζ > ¯

ζ−1as well as θ > 0and consider the candidate strategy Xθ,ζeκθ,0

with problem data (θ, ζeκθ,0). Then, for any ˜ϱ∈[−1,1], we obtain the lower bound

˜ϱ∇↑

tJθ(Xθ,ζeκθ,0) = w5(θ−t, ζ, ˜ϱ)

> w5(θ−t, ¯

ζ−1,−1) ≥0 (0 ≤t≤θ)(2.179)

due to monotonicity of the mapping w5in ζand ϱ. The last inequality in (2.179)

follows from 1.) with ϱ=−1. In particular, the desired property in (2.49) is

satisfied. Moreover, we have as in 1.) above that ˜ϱ∇↓

tJθ(ˆ

Xθ,ζeκθ,0)≥0for all t∈

[0, θ]. That is, the strategy ˆ

Xθ,ζeκθ,0is optimal by virtue of the first order optimality

conditions from Proposition 2.3.7. In particular, it holds that (θ, ζeκθ,0) ∈Rwait

by the definition of the waiting region in (2.24).

Proof of Corollary 2.3.26.As it was already the case for the proof of Corollary

2.3.24, the proof of Corollary 2.3.26 is again very similar to the proof of Corollary

2.3.21. Therefore, let us only summarize the main steps. Let ϑ > 0and let

Xθϱ+ϑ,ζ′,φ′denote the strategy from Corollary 2.3.26 with associated problem data

(θϱ+ϑ, ζ′, φ′).

First, we have to argue that ˆ

Xθϱ+ϑ,ζ′,φ′is optimal in view of the first order

optimality conditions from Proposition 2.3.7. Due to the dynamic programming

principle form Remark 2.3.9 as well as Lemma 2.3.8, we only have to check that

the buy-subgradient of ˆ

Xθϱ+ϑ,ζ′,φ′vanishes on [0, ϑ]. For convenience, as in the

proof of Corollary 2.3.21, we write φˆ

X,ζˆ

X,ˆ

X↑and ˆ

X↓instead of φˆ

Xθϱ+ϑ,ζ′,φ′,

ζˆ

Xθϱ+ϑ,ζ′,φ′,ˆ

Xθϱ+ϑ,ζ′,φ′,↑and ˆ

Xθϱ+ϑ,ζ′,φ′,↓, respectively. In fact, note that by the

2 Optimal Investment with Transient Price Impact

same computations as in the proof of Corollary 2.3.21 in step 1, it holds that

ϱ∇↑

tJθϱ+ϑ(ˆ

Xθϱ+ϑ,ζ′,φ′)

=λ2∫ϑ

φˆ

udu +1

2η∫ϑ

(e−κ(u−t)−eκ(u−t))dˆ

X↑

u−µ(ϑ−t)

2¯

ζϱ(eκ(θϱ+ϑ−t)−eκθϱ) (0 ≤t≤ϑ);

(2.180)

cf. the representation of the buy-subgradient in (2.162) above with τ= 0 and

φˆ

θϱ+ϑ= 0. Computing the first two integrals in (2.180) by using the explicit

definition of ˆ

Xθϱ+ϑ,ζ′,φ′as in Corollary 2.3.21 in (2.53) yields exactly the same

representation of the buy-subgradient as in (2.163) above with τ= 0,φˆ

θϱ+ϑ= 0

and function cas defined in (2.51) evaluated at (¯

ζϱeκθϱ,0) = (2µ/κ, 0). Specifically,

note that c(2µ/κ, 0) must satisfy

c(¯

ζϱeκθϱ,0) = β2

2λ2¯

ζeκθϱ=β2µ

λ2κ,

cf. also the corresponding equation in (2.164) above, which is indeed the case.

Moreover, using the representation of the buy-subgradient as in (2.163) (with τ= 0,

φˆ

θϱ+ϑ= 0,c(2µ/κ, 0)) as well as the fact that

ζϱeκθϱ=2µ

κ,

which is nothing but the second property in (2.70) of the optimal strategy ˆ

Xθϱ,¯

ζϱeκθϱ,0

from Proposition 2.3.25 (cf. the corresponding identity in (2.165)), one can deduce

that ϱ∇↑

tJθϱ+ϑ(ˆ

Xθϱ+ϑ,ζ′,φ′) = 0 for all 0≤t≤ϑas desired.

Finally, following the same line of computations as in the proof of Corollary

2.3.21 step 2, it can be shown that ˆ

Xθϱ+ϑ,ζ′,φ′is non-negative and increasing and

that ζ′≥¯

ζϱeκθϱ= 2µ/κ.

Proof of Proposition 2.3.27.The proof of Proposition 2.3.27 follows similar

arguments as in the proof of Proposition 2.3.20 2.).

1.): First, let ϱ∈(−1,1] be arbitrary but fixed and denote by θϱ,¯

ζϱthe corre-

sponding values from Proposition 2.3.25 1.). Observe that the map

w5(θ, ζ, ϱ) = −µθ +1

2ζ(eκθ +ϱ)(2.181)

from Lemma 2.4.3 3.) is strictly increasing in the third argument for ζ > 0. Hence,

together with the result from Proposition 2.3.25 1.), we obtain for any ¯ϱ∈[−1, ϱ)

the estimate

0 = ϱ∇↑

0Jθϱ(ˆ

Xθϱ,¯

ζϱeκθϱ,0) = w5(θϱ,¯

ζϱ, ϱ)> w5(θϱ,¯

ζϱ,¯ϱ).

2.4 Proofs

Consequently, since w5is continuous and strictly convex in its first argument (as

long as ζ > 0) with w5(0,¯

ζϱ,¯ϱ)>0, there must exist a unique θ∗

¯ϱ∈[0, θϱ)such that

0 = w5(θ∗

¯ϱ,¯

ζϱ,¯ϱ) = ¯ϱ∇↑

0Jθ∗

¯ϱ(ˆ

Xθ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0).

Moreover, again due to strict convexity of w5in the first argument and the fact

that ∂

∂θ w5is still strictly increasing in the third argument, we obtain together with

the results from Proposition 2.3.25 1.) the upper estimate

0 = d

dt (ϱ∇↑

tJθϱ(ˆ

Xθϱ,¯

ζϱeκθϱ,0))⏐⏐⏐⏐t↓0

=−∂

∂θw5(θϱ,¯

ζϱ, ϱ)

<−∂

∂θw5(θ∗

¯ϱ,¯

ζϱ, ϱ)<−∂

∂θw5(θ∗

¯ϱ,¯

ζϱ,¯ϱ)

dt (¯ϱ∇↑

tJθ∗

¯ϱ(Xθ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0))⏐⏐⏐⏐t↓0

That is, both properties in (2.74) are satisfied by Xθ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0. In addition, observe

that it follows by similar arguments as in the proof of Propposition 2.3.25 1.) that

the strategy Xθ∗

¯ϱ,¯

ζϱeκθ∗

¯ϱ,0is optimal by virtue of the first order optimality conditions

in Proposition 2.3.7.

2.): Finally, let ϱ= 1 be fixed and let ζ∈(0, e−κθ12µ/κ] = (0,¯

ζ1]be arbitrary but

fixed. In this case, we know from Proposition 2.3.25 2.) and thus from Proposition

2.3.23 1.) with θ=θ=θ1(and s2(θ) = ¯

ζ1) that

0 = 1∇↑

0Jθ(ˆ

Xθ,¯

ζ1eκθ,0) = w3(θ, ¯

ζ1,0,0) = w5(θ1,¯

ζ1,1) > w5(θ1, ζ, 1)

as well as

0 = d

dt (1∇↑

tJθ(ˆ

Xθ,¯

ζ1eκθ,0))⏐⏐⏐⏐t↓0

=−∂

∂θw3(θ, ¯

ζ1,0,0) = −∂

∂θw5(θ, ¯

ζ1,1)

>−∂

∂θw5(θ, ζ, 1).

Hence, observe that we can argue as in 1.).

Proof of Proposition 2.3.28.1.) Let us start with the assertion in 1.). There-

fore, let φ < 0be arbitrary and set ¯

ζ=s4(φ)as in (2.76). The verification of the

optimality of the strategy ˆ

Xϑ,ζ′,φ′with problem data (ϑ, ζ′, φ′)and arbitrary ϑ > 0

is again very close to the proof of Corollary 2.3.21 above. Hence, we only outline

the basic steps.

First, we have to argue that ˆ

Xϑ,ζ′,φ′satisfies the first order optimality condi-

tions from Proposition 2.3.7. Due to the dynamic programming principle form

Remark 2.3.9 and Lemma 2.3.8, we only have to check that the corresponding

buy-subgradient of ˆ

Xϑ,ζ′,φ′vanishes on [0, ϑ]. As in the proof of Corollary 2.3.21,

2 Optimal Investment with Transient Price Impact

we write φˆ

X,ζˆ

X,ˆ

X↑and ˆ

X↓instead of φˆ

Xθ+ϑ,ζ′,φ′,ζˆ

Xθ+ϑ,ζ′,φ′,ˆ

Xθ+ϑ,ζ′,φ′,↑and

Xθ+ϑ,ζ′,φ′,↓, respectively, to alleviate the notation. By the same computations as

in the proof of Corollary 2.3.21 in step 1, it holds that

∇↑

tJϑ(ˆ

Xϑ,ζ′,φ′)

=λ2∫ϑ

φˆ

udu +1

2η∫ϑ

(e−κ(u−t)−eκ(u−t))dˆ

X↑

u−µ(ϑ−t)

2¯

ζ(eκ(ϑ−t)−1) −1

2ηφ (e−κ(ϑ−t)−1)(0 ≤t≤ϑ);

(2.182)

cf. the representation of the buy-subgradient in (2.162) above with τ= 0,¯

θ= 0

and φˆ

ϑ=φ < 0. Computing the first two integrals in (2.182) by using the

explicit definition of ˆ

Xϑ,ζ′,φ′as given in Corollary 2.3.21 in (2.53) yields exactly

the same representation of the buy-subgradient as in (2.163) above with τ=¯

θ= 0,

φˆ

ϑ=φ < 0and function cas defined in (2.51) evaluated at (¯

ζ, φ). Specifically,

note that c(¯

ζ, φ)must satisfy

c(¯

ζ, φ) = β2

2λ2(¯

ζ−ηφ).(2.183)

Indeed, inserting the definition of ¯

ζfrom (2.76) in (2.183) yields

c(¯

ζ, φ) = β2

λ2(¯

ζ+1

κ(λ2φ−µ))

as desired. In addition, employing further the definition of ¯

ζin (2.76), one can

deduce via elementary algebraic manipulations as in the proof of Corollary 2.3.21

step 1 above that ∇↑

tJϑ(ˆ

Xϑ,ζ′,φ′) = 0 actually holds true for all 0≤t≤ϑ.

Finally, following the same line of computations as in the proof of Corollary

2.3.21 step 2, it can be shown that ˆ

Xϑ,ζ′,φ′is non-negative and increasing and that

ζ′≥¯

ζ.

2.) Now, let ζ > ¯

ζbe arbitrary but fixed and consider the candidate strategy

Xθ,ζeκθ,φ in (2.75) with problem data (θ, ζeκθ, φ)where θ > 0is arbitrary. We have

to argue via the first order optimality conditions from Proposition 2.3.7 that the

candidate strategy Xθ,ζeκθ,φ ∈Xdis optimal and satisfies the desired property in

(2.78). Note that the optimal strategy ˆ

X0,ζ,φ with data (0, ζ, φ)and φ < 0satisfies

∇↑

0J0(ˆ

X0,ζ,φ) = 0 (cf. also Remark 2.3.16). Thus, by virtue of Lemma 2.4.3 1.)

with τ= 0, the sell-subgradient on [0, θ]is given by

∇↓

tJθ(Xθ,ζeκθ,φ) = ∇↓

0Jθ−t(Xθ−t,ζeκ(θ−t),φ) = w1(θ−t, ζ, φ, 0)

= (µ−λ2φ)(θ−t) + 1

2ζ(eκ(θ−t)+ 1) −1

2ηφ(e−κ(θ−t)+ 1) >0.(2.184)

Concerning the buy-subgradient on [0, θ]of our candidate strategy ˆ

Xθ,ζeκθ,φ in

(2.75), we obtain by Lemma 2.4.3 1.) the expression

∇↑

tJθ(Xθ,ζeκθ,φ) = ∇↑

0Jθ−t(Xθ−t,ζeκ(θ−t),φ) = w2(θ−t, ζ, φ, 0)

=−(µ−λ2φ)(θ−t) + 1

2ζ(eκ(θ−t)−1) −1

2ηφ(e−κ(θ−t)−1)

2.4 Proofs

as well as the upper bound

dt∇↑

tJθ(Xθ,ζeκθ,φ) = −∂

∂θw2(θ−t, ζ, φ, 0)

=µ−λ2φ−1

2κζeκ(θ−t)−1

2κηφe−κ(θ−t)<0,

(2.185)

since φ < 0and ζ > ¯

ζ= (2µ−φ(2λ2+ηκ))/κ. As a consequence, since θ↦→

w2(θ, ζ, φ, 0) is strictly convex with w2(0, ζ, φ, 0) = ∂

∂θ w2(0, ζ, φ, 0) = 0, we have

∇↑

tJθ(Xθ,ζeκθ,φ)>0 (0 ≤t < θ).(2.186)

Thus, the desired property in (2.78) is satisfied. In particular, in view of the first

order optimality conditions from Proposition 2.3.7, the inequalities in (2.184) and

(2.186) imply the optimality of Xθ,ζeκθ,φ with problem data (θ, ζeκθ, φ)∈Rwait.

3 High Resilience Heuristics for Utility-Based Hedging

In this chapter we want to discuss the problem of hedging a European contingent

claim in our illiquid financial market model presented in Chapter 2, Section 2.1,

above. Specifically, let us consider an agent who wants to hedge a European-type

option with payoff Hat maturity Tbut faces liquidity costs LT(X)as introduced

in (2.6) which are levied on her trading activities X= (X↑, X↓)∈Xdue to market

tightness ζ > 0, finite market depth η > 0and finite resilience κ > 0. In such a

model with illiquidity frictions, one economically appealing choice for an agent

who has to manage the financial risk incurred by a short position in some claim H

is given by the well-known utility-indifference pricing and hedging approach put

forward by Hodges and Neuberger (1989) as well as Davis et al. (1993): For a given

preference structure of the agent described by a utility function u:R→R, her

aim is to find a p∈Rsuch that

max

X=(X↑,X↓)∈X

E[u(VT(X)−H+p)] = max

X=(X↑,X↓)∈X

E[u(VT(X))] (3.1)

holds true. Recall from Section 2.1 that VT(X) = ξX

0−+φX

0−P0−+∫T

0φX

tdPt−LT(X)

denotes her final liquidation wealth given by (2.5) with initial frictionless wealth

ξX

0−+φX

0−P0−. The process (Pt)t≥0represents the unaffected price process in the

bid- and ask-price dynamics in (2.1) and is modelled by a continuous semimartin-

gale. The difference of the two optimal investment strategies in (3.1) with and

without random endowment −H+pis then considered as the agent’s so-called

utility-based hedge against the financial risk of writing the claim H. As mentioned

in Remark 2.2.4 2.) above, within the setup of Section 2.2, the existence proof

of Theorem 2.2.3 for an optimal strategy of the utility maximization problem also

works in the presence of, e.g., a bounded initial random endowment H∈FT.

Our case study in Section 2.3 revealed that in an illiquid Bachelier market

the problem of maximizing expected exponential utility from terminal liquidation

wealth as formulated in (2.11) without claim reduces to the deterministic optimal

tracking problem in (2.13). In fact, in case where the random claim Hexhibits

a deterministic Bachelier delta hedge, the resulting utility maximization problem

with additional endowment still admits a deterministic optimizer.

Lemma 3.0.1. Assume that the unaffected price process (Pt)t≥0is a P-Brownian

motion with drift µ= 0, volatility σ > 0and that the claim H∈L∞(FT,P)is

given by

H=E[H] + ∫T

ξH

tdPt

3 High Resilience Heuristics for Utility-Based Hedging

with deterministic process (ξH

t)0≤t≤T. Then, the optimal strategy of the utility

maximization problem in (2.11)with random endowment Hover the set of bounded

strategies X∈¯

Xis deterministic and coincides with the minimizer of the convex

cost functional

LT(X) + ασ2

2∫T

0(ξH

t−φX

t)2dt. (3.2)

Proof. The claim follows by exactly the same argument as in the proof of Proposi-

tion 2.3.2 in Section 2.3 above.

Note that analogously to the optimal portfolio choice problem without claim

in (2.13), the cost functional in (3.2) leads to an optimal tracking problem whereas

the traget strategy is now given by the deterministic delta-hedge ξHinstead of the

frictionless constant Merton portfolio.

Obviously, in general, the presence of a random endowment Hin the utility

maximization problem in the illiquid Bachelier model from Section 2.3 will rule out

optimality of deterministic strategies. Moreover, in view of the involved case study

presented in Section 2.3 concerning the classical Merton problem without claim,

it is very sensible to expect that one cannot hope for explicit solutions of optimal

strategies for the utility maximization problem with random endowment. Put

differently, obtaining explicit characterizations of utility-based hedging strategies

of contingent claims in our price impact model from Section 2.1 appears to be

unachievable already in the most elementary setting.

One possible remedy consists of resorting to suitable asymptotic expansions.

More precisely, motivated by the ideas in Roch and Soner (2013), Kallsen and

Muhle-Karbe (2014), Moreau et al. (2017) or Ekren and Muhle-Karbe (2017), we

will consider the utility-based hedging problem in our price impact model from

Section 2.1 asymptotically in the “high resilience limit” when the resilience param-

eter κ, i.e., the recovery of the bid- and ask-prices in (2.1), converges to infinity.

Henceforth, we want to explain heuristically how the utility maximization problem

with random endowment Hin our illiquid market model can be related asymp-

totically for κ↑ ∞ to a considerably simpler linear quadratic optimal hedging

problem. We point out that we are just loosely drawing a connection between

these two problems. We do neither provide rigorous convergence statements nor

proofs. The obtained simpler stochastic linear quadratic benchmark problem is

then comprehensively studied in the next Chapters 4and 5.

To fix ideas, let us place ourselves in the setup of Section 2.1. We start with

the observation that in the limit for κ↑ ∞ our price impact model introduced in

Section 2.1 reduces to a model with permanent price impact in the following sense.

Lemma 3.0.2. For a fixed strategy X∈Xwith φX

T= 0 and [φX]T= 0, the

terminal liquidation value Vκ

T(X)≜VT(X)as given in (2.5)with liquidity costs

Lκ

T(X)≜LT(X)as given in (2.6)converges for κ↑ ∞ to the P&L in a model with

permanent price impact and permanent impact parameter η/2:

lim

κ→∞Vκ

T(X)≜V∞

T(X)≜ξX

0−+φX

0−P0−+∫[0,T ]

φX

t−(dPt+η

2dφX

=ξX

0−+φX

0−P0−−η

4(φX

0−)2+∫[0,T ]

φX

t−dPt

(3.3)

pointwise on Ω.

Proof. Using the representation of the liquidity costs LT(X)in (2.86), we obtain

under the above assumptions on the fixed strategy X∈Xthe expression

Lκ

T(X) = η

4(φX

0−)2+1

2∫[0,T ]

ζX

t−(dX↑

t+dX↓

t).(3.4)

Moreover, since φX

T= 0, applying integration by parts yields the identity

2∫[0,T ]

φX

t−dφX

t=−η

4(φX

0−)2

and thus

Vκ

T(X) = ξX

0−+φX

0−P0−+∫[0,T ]

φX

t−(dPt+η

2dφX

t)−1

2∫[0,T ]

ζX

t−(dX↑

t+dX↓

t).

Finally, as ζX

t↓0for κ↑ ∞ pointwise on Ω, monotone convergence yields the claim

in (3.3).

Lemma 3.0.2 shows that, loosely speaking, for a given bounded contingent claim

H∈FT, the utility maximization problem

E[u(V∞

T(X)−H+p)] →max (3.5)

with random endowment −H+pin the limiting model with terminal wealth V∞

T(X)

as in (3.3) (over a properly chosen set of admissible share holdings X) amounts

to a frictionless utility maximization problem with claim Hand adjusted initial

frictionless wealth v∞

0≜ξX

0−+φX

0−P0−−η

4(φX

0−)2. We denote by X∞the optimal

share holdings of the corresponding frictionless optimizer in (3.5) which we assume

to exist; cf., e.g., Hugonnier and Kramkov (2004) for related results. Then, if

the resilience parameter κis very large, one can expect that the optimal value

E[u(Vκ

T(X)−H+p)] of the original control problem over, e.g., the set of strategies

X∈Xwith same initial frictionless wealth v∞

0satisfying φX

T= [φX]T= 0, is close

to the frictionless optimal value E[u(V∞

T(X∞)−H+p)]. Therefore, adopting the

approach from Kallsen and Li (2013) with exponential utility function

u(x) = −e−αx, α > 0,

3 High Resilience Heuristics for Utility-Based Hedging

we may approximate, up to second order quantities, the difference via

E[u(V∞

T(X∞)−H+p)] −E[u(Vκ

T(X)−H+p)] (3.6)

≃E[u′(V∞

T(X∞)−H+p)(V∞

T(X∞)−Vκ

T(X))

−1

2u′′(V∞

T(X∞)−H+p)(V∞

T(X∞)−Vκ

T(X))2]

≃E∞[Lκ

T(X) + α

2(V∞

T(X∞)−Vκ

T(X))2]E[u′(V∞

T(X∞)−H+p)]

≃E∞⎡

⎣Lκ

T(X) + α

2(∫T

0(X∞

t−φX

t)dPt)2⎤

⎦E[u′(V∞

T(X∞)−H+p)]

≃E∞[Lκ

T(X) + α

2∫T

0(X∞

t−φX

t)2d⟨P⟩t]E[u′(V∞

T(X∞)−H+p)](3.7)

where E∞denotes an integration with respect to the martingale measure P∞with

density

dP∞

dP=u′(V∞

T(X∞)−H+p)

E[u′(V∞

T(X∞)−H+p)]

for the unaffected price process P; again, cf., e.g., Hugonnier and Kramkov (2004).

Moreover, at least for smooth strategies X, we may push the derived approxima-

tion in (3.7) even further by additionally approximating the liquidity costs Lκ

T(X)

in (3.7). That is, using their representation in (3.4), the spread dynamics as given

in (2.3) as well as the notation ∥X∥≜X↑+X↓, we may approximate, assuming

for simplicity φX

0−=ζX

0−= 0, the liquidity costs via

Lκ

T(X) = η

4(φX

0−)2+1

2∫[0,T ]

ζX

t−d∥X∥t

=η

2∫[0,T ](∫t

e−κ(t−u)∥˙

X∥udu)d∥X∥t

=η

2κ∫[0,T ](∫t

κe−κτ ∥˙

X∥t−τdτ)d∥X∥t

≃η

2κ∫[0,T ]∥˙

X∥td∥X∥t=η

2κ∫T

0(˙φX

t)2dt, (3.8)

because, for κ“large”, we may simply write

∫t

κe−κτ ∥˙

X∥t−τdτ ∼

κ↑∞ ∥˙

X∥t.

As a consequence, inserting the approximation in (3.8) back into (3.7), finding an

absolutely continuous strategy Xwhich minimizes the difference in (3.6) above

is heuristically under a “high resilience” regime equivalent to the simplified linear

quadratic stochastic optimization problem

E∞[η

2κ∫T

0(˙φX

t)2dt +α

2∫T

0(X∞

t−φX

t)2d⟨P⟩t].(3.9)

Observe that the benchmark problem in (3.9) can be regarded as an optimal tracking

problem of the frictionless optimal utility-based hedge X∞from the limiting model

in the presence of quadratic transaction costs; cf. also a related discussion in Cai

et al. (2015, Section 5). The optimal strategy seeks to minimize both the squared

deviation of its share holdings φXfrom the frictionless target strategy X∞as well

as the incurred quadratic trading costs which are levied on the respective turnover-

rates ˙φX. In particular, these costs can be interpreted as being induced by linear

temporary price impact as in Almgren and Chriss (2001). In other words, similar

to the idea presented in Kallsen and Muhle-Karbe (2014), the utility maximization

problem with random endowment in our original Obizhaeva and Wang (2013) type

model from Section 2.1 can be reduced, at least heuristically, under “high resilience”

to a simpler linear quadratic problem in the Almgren and Chriss (2001) setup with

“small” temporary price impact. This reduction of the original hedging problem to

the optimal tracking problem in (3.9) allows the option seller to choose a preferred,

yet due to temporary price impact not implementable hedging strategy adopted

from a frictionless setting as her traget strategy which she seeks to track in order

to manage her financial risk of writing the claim. This motivates our study of

linear quadratic stochastic control problems of the above type (3.9) in the next two

chapters.

Remark 3.0.3.1. Note that in the particular situation of Lemma 3.0.1 the mini-

mizer of the difference in (3.6) actually coincides with the minimizer of (3.7).

2. In Chapter 4the quotient η/(2κ)in the linear quadratic control problem

in (3.9) will be replaced by κ; cf. the considered objective functional in (4.2)

below. That is, the parameter κwill represent henceforth temporary price

impact rather than the resilience rate.

4 Hedging with Temporary Price Impact

This chapter is joint work with Peter Bank and H. Mete Soner and published in the

journal Mathematics and Financial Economics; cf. Bank, Soner, and Voß (2017).

The final publication is available at Springer via http://dx.doi.org/10.1007/s11579-

016-0178-4.

We consider the problem of hedging a European contingent claim in a Bache-

lier model with temporary price impact as proposed by Almgren and Chriss (2001).

Motivated by our heuristic considerations from Chapter 3, the hedging problem can

be regarded as a cost optimal tracking problem of the frictionless hedging strategy.

We solve this problem explicitly for general predictable target hedging strategies in

Section 4.2. It turns out that, rather than towards the current target position, the

optimal policy trades towards a weighted average of expected future target posi-

tions. This generalizes an observation of Gârleanu and Pedersen (2013b) from their

homogenous Markovian optimal investment problem to a general hedging problem.

Our findings complement a number of previous studies in the literature on optimal

strategies in illiquid markets as, e.g., Gârleanu and Pedersen (2013b), Naujokat

and Westray (2011), Rogers and Singh (2010), Almgren and Li (2016), Moreau

et al. (2017), Kallsen and Muhle-Karbe (2014), Guasoni and Weber (2017,2015),

where the frictionless hedging strategy is confined to diffusions. The consideration

of general predictable reference strategies is made possible by the use of a convex

analysis approach instead of the more common dynamic programming methods.

4.1 Problem setup

We fix a finite deterministic time horizon T > 0, a filtered probability space (Ω,F,

(Ft)0≤t≤T,P)satisfying the usual conditions of right continuity and completeness

and consider an agent who is trading in a financial market consisting of a risky

asset, e.g., stock. The number of shares the agent holds at time t∈[0, T]of the

risky stock is defined as

t≜x+∫t

usds, 0≤t≤T, (4.1)

where x∈Rdenotes her given initial holdings. The real-valued stochastic process

(ut)0≤t≤Trepresents the agent’s turnover rate, that is, the speed at which the agent

trades in the risky asset. It is assumed to be chosen in the set

U≜{u:uprogressively measurable s.t. E∫T

tdt < ∞}.

4 Hedging with Temporary Price Impact

The square-integrability requirement ensures that the induced quadratic transac-

tion costs which are levied on the agent’s respective turnover rates due to temporary

price impact as in Almgren and Chriss (2001) are finite.

In such a frictional market, our agent seeks to track a target strategy which

can be thought of, for instance, as a hedging strategy adopted from a frictionless

setting. Mathematically, this problem can be formalized as follows: Given a real-

valued predictable process (ξt)0≤t≤Tin L2(P⊗dt)and a fixed constant κ > 0, the

agent’s objective is to minimize the performance functional

J(u)≜E[1

2∫T

(Xu

t−ξt)2dt +1

2κ∫T

tdt].(4.2)

This leads to the optimal stochastic control problem

J(u)→min

u∈U.(4.3)

Since the agent’s terminal position Xu

Tmay be important (for here future plans or

physical delivery), we also consider the optimal stochastic control problem

J(u)→min

u∈UΞ

(4.4)

where UΞ

xis the set of constrained policies defined as

UΞ

x≜{u:u∈Usatisfying Xu

T≡x+∫T

usds = ΞTP-a.s.}

with predetermined terminal position ΞT∈L2(P,FT)such that

∫T

dE[Ξ2

T−t<∞(4.5)

where Ξt≜E[ΞT|Ft]for 0≤t≤T.

Remark 4.1.1.1. Lemma 4.4.4 below shows that a target ΞTcan be reached

with finite expected costs in the sense that UΞ

x=∅if and only if (4.5) is

satisfied. Observe that this condition implies, in particular, that ΞT∈FT−.

In fact, (4.5) can be interpreted as a condition on the speed at which one

learns about the ultimate target position ΞTas t↑T.

2. Concerning physical delivery at maturity T, it would be sufficient to impose

the constraint Xu

T≥ΞT. However, this would lead to an interesting, yet tech-

nically rather different optimization problem which is left for future research.

Recall that our motivation of the objective functional in (4.2) and its connection

to the problem of hedging a European contingent claim in the presence of temporary

price impact is presented in the preceding Chapter 3. More precisely, it is a variant

of the linear quadratic functional formulated in (3.9) where the unaffected price

4.2 Main results

process is given by a Brownian motion. The general case allowing for stochastic

temporary price impact and stochastic volatility in the cost functional (4.2) is

studied in the next Chapter 5.

Remark 4.1.2.

1. A similar hedging problem as formulated in (4.2) is also studied in Rogers

and Singh (2010) and Almgren and Li (2016). In contrast to our setting,

Rogers and Singh (2010) consider a Black-Scholes framework. Almgren and

Li (2016) also include permanent price impact.

2. Apart from hedging, the minimization problem of the objective in (4.2) is also

related to the problem of optimally executing a VWAP order as studied using

dynamic programming methods in a Markovian setup in Frei and Westray

(2013) and Cartea and Jaimungal (2016), or, more generally, to the optimal

curve following problem as discussed in Naujokat and Westray (2011) as well

as Cai et al. (2015).

3. In a Brownian setting, our problem (4.3) is a special case of a stochastic linear

quadratic control problem as studied, e.g., by Kohlmann and Tang (2002);

cf. also the more general framework discussed in Chapter 5below.

4.2 Main results

Our main results are the following explicit descriptions of the optimal controls

for problems (4.3) and (4.4) and their corresponding minimal costs for which it is

convenient to introduce

τκ(t)≜T−t

√κ,0≤t≤T.

Theorem 4.2.1. The optimal stock holdings ˆ

Xof problem (4.3) with unconstrained

terminal position satisfy the linear ODE

dˆ

Xt=tanh(τκ(t))

√κ(ˆ

ξt−ˆ

Xt)dt, ˆ

X0=x, (4.6)

where, for 0≤t < T, we let

ξt≜E[∫T

ξuK(t, u)du⏐⏐⏐⏐

Ft](4.7)

with the kernel

K(t, u)≜cosh(τκ(u))

√κsinh(τκ(t)),0≤t≤u < T.

4 Hedging with Temporary Price Impact

The minimal costs are given by

inf

u∈UJ(u) =1

2√κtanh(τκ(0))(x−ˆ

ξ0)2+1

2E[∫T

(ξt−ˆ

ξt)2dt]

2E[∫T

√κtanh(τκ(t))d⟨ˆ

ξ⟩t]<∞.(4.8)

For the constrained problem we have similarly:

Theorem 4.2.2. The optimal stock holdings ˆ

XΞof problem (4.4) with constrained

terminal position ΞT∈L2(P,FT)such that (4.5)holds satisfy the linear ODE

dˆ

XΞ

t=coth(τκ(t))

√κ(ˆ

ξΞ

t−ˆ

XΞ

t)dt, ˆ

XΞ

0=x, (4.9)

where, for 0≤t≤T, we let

ξΞ

t≜E[1

cosh(τκ(t))ΞT+(1−1

cosh(τκ(t)))∫T

ξuKΞ(t, u)du⏐⏐⏐⏐

Ft],(4.10)

with the kernel

KΞ(t, u)≜sinh(τκ(u))

√κ(cosh(τκ(t)) −1),0≤t≤u < T.

The solution ˆ

XΞof (4.9)satisfies the terminal constraint in the sense that

lim

t↑T

XΞ

t= ΞTP-a.s.

The minimal costs are given by

inf

u∈UΞJ(u) =1

2√κcoth(τκ(0))(x−ˆ

ξΞ

0)2+1

2E[∫T

(ξt−ˆ

ξΞ

t)2dt]

2E[∫T

√κcoth(τκ(t))d⟨ˆ

ξΞ⟩t]<∞.(4.11)

The convex-analytic proofs of Theorems 4.2.1 and 4.2.2 are deferred to Sec-

tion 4.4. Note that, rather than towards the current target position ξt, the optimal

frictional hedging rules in (4.6) and (4.9) prescribe to trade towards weighted av-

erages ˆ

ξtand ˆ

ξΞ

t, respectively, of expected future target positions of ξ. Indeed, for

each 0≤t≤T,K(t, .)and KΞ(t, .)specify nonnegative kernels which integrate

to one over [t, T], and so ˆ

ξand ˆ

ξΞaverage out the expected future positions of

ξ. For ˆ

ξΞone chooses a convex combination of this average of ξwith the ex-

pected terminal position ΞT, where the weight shifts gradually to ΞTas t↑Tsince

1/cosh(τk(t)) ↑1in that case.

According to (4.6) and (4.9), the optimal tracking rate trades towards these tar-

gets at a speed proportional to their distance to the investor’s position at any time.

4.2 Main results

The coefficient of proportionality is controlled by both the cost parameter κand

the remaining time-to-maturity T−t. For the unconstrained solution in (4.6), since

limt↑Ttanh(τκ(t)) = 0, trading slows down when approaching the final time T; in

other words, towards the end, the investor does not worry about tracking ξany-

more, but seeks to minimize trading costs. This becomes intuitive when comparing

the effect of early interventions to later ones: with early interventions the investor

ensures that she stays reasonably close to the target for the foreseeable future, but

late interventions only can impact the investor’s performance for very short periods

and therefore do not warrant, at least asymptotically, the associated costs. For the

constrained solution in (4.9) by contrast, we have limt↑Tcoth(τκ(t)) = +∞and

so the optimal strategy trades with increased urgency towards ˆ

ξΞ, which itself is

easily seen to converge to the ultimate target position ΞT= limt↑Tˆ

ξΞ

tP-a.s. (cf.

Proof of Theorem 4.2.2 below in Section 4.4).

Our result generalizes an observation from Gârleanu and Pedersen (2013b) from

their homogeneous Markovian optimal investment problem to a general hedging

problem with a general predictable target strategy ξ, also allowing for a random

terminal portfolio position ΞT. It also sheds further light on the general structure

of optimal portfolio strategies in markets with frictions. Indeed, the description

of (asymptotically) optimal trading strategies obtained in Moreau et al. (2017),

Kallsen and Muhle-Karbe (2014), or Guasoni and Weber (2017,2015) prescribe a

reversion towards the frictionless strategy ξitself, not towards an average such as ˆ

or ˆ

ξΞ. For sufficiently smooth ξ, e.g., of diffusion type, this is still optimal asymp-

totically for small liquidity costs as then these averages do not differ significantly

from ξ. The next section, however, shows that this is no longer the case when we

allow for singularities in the reference strategy.

Finally, our representations (4.8) and (4.11) for the values of the tracking prob-

lems (4.3) and (4.4), respectively, show how these depend on the initial position x

and the L2-distance between the target ξand the respective signal processes ˆ

ξand

ξΞ. It also reveals the importance of the signals’ quadratic variation ⟨ˆ

ξ⟩,⟨ˆ

ξΞ⟩which

can be viewed as a measure for how effectively one can predict the target positions

ξand ΞT. To the best of our knowledge, the key role played by the signals ˆ

ξ,ˆ

ξΞwas

not observed in the general theory of stochastic linear-quadratic control problems

as discussed, e.g., by Kohlmann and Tang (2002). In fact, this is preserved when

passing to a more general setup as we will do so in Chapter 5below.

Remark 4.2.3.As mentioned in the description of our problem setup in Section 4.1,

the quadratic cost term in our objective function in (4.2) is due to linear temporary

price impact as in the model proposed by Almgren and Chriss (2001). In this

regard, one might likewise extend the objective functional also in order to account

for expected costs resulting from linear permanent price impact as in Almgren and

4 Hedging with Temporary Price Impact

Chriss (2001). This would lead to the introduction of the additional term

E⎡

⎣θ(∫T

utdt)2⎤

⎦=θE[(Xu

T−x)2](4.12)

for some constant θ > 0. For the constrained problem in (4.4), this extra cost

term obviously does not depend on the strategy and is thus irrelevant. For the

unconstrained problem in (4.3), these extra costs can be regarded as a penalization

term forcing the final position Xu

Tto be close to the initial position x. For the ease

of exposition, we refrain in the present paper from inducing this additional term,

since our main intention here is to consider the simplest case and to outline the key

role played by the optimal tracking signals ˆ

ξ,ˆ

ξΞ, respectively, in the description

of the optimal control as well as the corresponding minimal costs. A more general

setup allowing for stochastic price impact, stochastic volatility and a stochastic

penalization on the terminal position as in (4.12) is considered in Chapter 5.

4.3 Illustrations

In this section we present a few case studies illustrating the structure of the optimal

hedging strategies we found in Theorems 4.2.1 and 4.2.2. The first two case studies

are simple deterministic toy examples which allow us to understand the effect of

jumps as well as of initial and terminal positions. The final case study considers

a discretely monitored Asian option where random jumps in the reference hedge

occur naturally.

In the first two cases we assume the initial position to be x= 0 and consider

a time horizon of T= 1 when, in the constrained case, the position has to be

liquidated, i.e., ΞT= 0. We depict ξalong with its averages ˆ

ξand ˆ

ξΞ, respectively,

as well as the corresponding optimal frictional hedges ˆ

Xand ˆ

XΞ. We also include a

“myopic” benchmark strategy ˜

Xwhich directly targets ξ(without final constraint)

given by

d˜

Xt=1

√κ(ξt−˜

Xt)dt, 0≤t≤T,

in order to compare with analogous strategies considered in Rogers and Singh

(2010), Moreau et al. (2017), Guasoni and Weber (2017,2015) and Kallsen and

Muhle-Karbe (2014).

4.3.1 Frictionless deterministic hedge with a jump

In our first case study we consider a deterministic target strategy ξ(solid blue line

in Figure 4.1) which can be viewed as a stock-buying schedule that prescribes to

hold one stock until time T/2when the position is doubled by a jump.

One can observe that the effective target strategies ˆ

ξand ˆ

ξΞof the optimal

controls ˆuand ˆuΞ, respectively, are smoothing out the jump of ξ. The target ˆ

ξΞ

4.3 Illustrations

additionally takes into account the liquidation constraint ΞT= 0 of the agent’s

position until maturity T. As expected, the optimal frictional hedges ˆ

Xand ˆ

XΞ

are indeed anticipating the upward jump of the target strategy ξat t=T/2by

building up their positions beyond the actual current position of ξeven before the

occurrence of the jump. This is not the case for the myopic benchmark strategy

Xwhich increases its position much more slowly and exhibits a kink when the

jump occurs after which trading speed picks up significantly. Finally, the optimal

holdings ˆ

XΞin the constrained setting, where the position has to be unwound

ultimately, are decreasing when time approaches maturity and end up in the final

desired position ˆ

XΞ

T= 0.

0.0 0.2 0.4 0.6 0.8 1.0

time0.0

0.5

1.0

1.5

2.0

2.5

3.0

number of shares

Figure 4.1: Frictionless hedge ξwith a jump at t=T/2(blue), corresponding

unconstrained (orange, dashed) and constrained (green, dashed) targets

ξand ˆ

ξΞ, respectively, as well as the corresponding frictional hedges ˆ

(orange line) and ˆ

XΞ(green line). The myopic benchmark hedge ˜

Xis

plotted in red.

4.3.2 Frictionless deterministic hedge with a singularity

The second target strategy ξ(solid blue line in Figure 4.2) is again deterministic

and also exhibits a singularity midway at t=T/2, this time, however, it is a jump

from −∞ to +∞.

Once more, one can observe that the effective target strategies ˆ

ξand ˆ

ξΞof the

optimal controls ˆuand ˆuΞ, respectively, are smoothing out the singularity of ξ.

Again, the target ˆ

ξΞadditionally takes into account the liquidation constraint ΞT=

0of the agent’s position until maturity T. In contrast to the benchmark strategy

X, the optimal frictional hedges ˆ

Xand ˆ

XΞare anticipating the singularity of the

target strategy ξat t=T/2by gradually building up their positions before the

4 Hedging with Temporary Price Impact

singularity occurs. Actually, they are trading away from the current target positions

of ξfor some time prior to T/2. This is in stark contrast with the myopic benchmark

strategy which keeps selling short more and more intensely even milliseconds before

the reference strategy jumps to +∞.

0.2 0.4 0.6 0.8

1.0

time

-4

-2

number of shares

Figure 4.2: Frictionless hedge ξwith a singularity at t=T/2(blue), corresponding

unconstrained (orange, dashed) and constrained (green, dashed) targets

ξand ˆ

ξΞ, respectively, as well as the corresponding frictional hedges ˆ

(orange line) and ˆ

XΞ(green line). The myopic benchmark hedge ˜

Xis

plotted in red.

4.3.3 Discrete Asian option

In this final example we investigate a situation where the target strategy ξis

stochastic and exhibits a random jump. Specifically, we consider hedging a dis-

crete Asian call with maturity T > 0in the Bachelier model where the underlying

risky asset Sis modeled by a Brownian motion with volatility σ > 0:

St=S0+σWt,0≤t≤T.

For simplicity, we assume that the average is discretely monitored over two fixing

dates T/2and T. That is, the payoff at maturity Tis given by

H≜(1

2(ST/2+ST)−K)+

for some strike K∈R. The Bachelier price of the discrete Asian option at time

t∈[0, T)can be computed as

πt≜

⎧

⎪

⎨

⎪

⎩

σ√5T/8−t φ(St−K

σ√5T/8−t)+StΦ(St−K

σ√5T/8−t),0≤t < T/2

2σ√T−t φ(ST/2+St−2K

σ√T−t)+(1

2(ST/2+St)−K)Φ(ST/2+St−2K

σ√T−t), T/2≤t < T

4.3 Illustrations

0.0 0.2 0.4 0.6 0.8 1.0

0.5

time

number of shares

moneyness

Figure 4.3: Frictionless hedge ξwith a jump at t=T/2(blue), corresponding

unconstrained (orange, dashed) and constrained (green, dashed) targets

ξand ˆ

ξΞ, respectively, as well as the corresponding frictional hedges ˆ

(orange line) and ˆ

XΞ(green line). The myopic benchmark hedge ˜

Xis

plotted in red. The moneyness is indicated by the light gray line.

where φand Φdenote the density and the cumulative distribution function of

the standard normal distribution, respectively. Accordingly, the frictionless delta-

hedging strategy is

ξt=⎧

⎪

⎨

⎪

⎩

Φ(St−K

σ√5T/8−t),0≤t≤T/2

2Φ(ST/2+St−2K

σ√T−t), T/2< t < T.

Note that the delta-hedge exhibits a negative random jump at time T/2since

ξT

2+−ξT

2−≜lim

t↓T

ξt−lim

t↑T

ξt=−1

2Φ(ST/2−K

σ√T/8).

We assume that the initial position xcoincides with the initial frictionless delta,

i.e., e.g., x= 1/2in the case of an at-the-money option with K=S0. This allows

us to focus on the hedging performance itself and avoids distortions from the initial

built up of a sensible hedging position. As before, the terminal position will be

allowed to be either unconstrained or mandating liquidation, i.e., ΞT= 0.

The effective targets ˆ

ξand ˆ

ξΞof the optimal frictional hedging strategy in (4.6)

4 Hedging with Temporary Price Impact

and (4.9), respectively, can be explicitly computed:

ξt=⎧

⎨

⎩

Φ(2(St−K)

σ√5T/2−4t)(1−1

sinh(τκ(T/2))

sinh(τκ(t)) ),0≤t < T/2,

ξt, T/2≤t < T,

and

ξΞ

t=⎧

⎪

⎨

⎪

⎩

Φ(2(St−K)

σ√5T/2−4t)(1−1

cosh(τκ(T/2))+1

cosh(τκ(t)) ),0≤t < T/2

(1−1

cosh(τκ(t)) )ξt, T/2≤t < T.

Observe that the Bachelier delta-hedge ξis a martingale on [T/2, T]and thus

the signal ˆ

ξcoincides with it in thi period. However, the optimal target ˆ

ξdiffers

from the frictionless hedge ξon [0, T/2] since it is anticipating and systematically

smoothing out the random jump at T/2whose size is determined by the option’s

moneyness at this point. The constrained target ˆ

ξΞanticipates the liquidation re-

quirement at maturity which plays a more and more dominating role after time T/2.

Again, the myopic benchmark strategy

d˜

Xt=σ

√κ(ξt−˜

Xt)dt, 0≤t < T

is not taking into account the random jump at time T/2and keeps on tracking the

frictionless delta-hedge even milliseconds before T/2(see Figure 4.3).

4.4 Proofs

In order to prove our main Theorems 4.2.1 and 4.2.2 we use tools from convex

analysis. Note that the performance functional u↦→ J(u)in (4.2) is strictly convex.

Given a control u∈Urecall the definition of the Gâteaux derivative of Jat uin

the direction of w∈L2(P⊗dt):

⟨J′(u), w⟩≜lim

ρ→0

J(u+ρw)−J(u)

ρ.

The following lemma provides an explicit expression for the Gâteaux derivative of

our performance functional J:

Lemma 4.4.1. For u∈Uwe have

⟨J′(u), w⟩=E[∫T

ws(κus+∫T

(Xu

t−ξt)dt)ds]

for any w∈L2(P⊗dt).

Proof. Let ρ > 0,u∈Uand w∈L2(P⊗dt). Note that Xu+ρw

t=Xu

t+ρ∫t

0wsds.

Then, we have

J(u+ρw)−J(u) =ρE[∫T

κutwt+(∫t

wsds)(Xu

t−ξt)dt]

+ρ2E[κ

2∫T

tdt +1

2∫T

0(∫t

wsds)2

dt].

100

4.4 Proofs

Hence,

⟨J′(u), w⟩=E[∫T

κutwt+(∫t

wsds)(Xu

t−ξt)dt].

Note that due to Fubini’s Theorem we can write the second part of the above

integral as

∫T

0(∫t

wsds)(Xu

t−ξt)dt =∫T

0(∫T

(Xu

t−ξt)dt)wsds

which finally yields the assertion.

Let us next derive necessary and sufficient first order conditions for problems

(4.3) and (4.4).

Lemma 4.4.2 (First order conditions).

1. In the unconstrained problem (4.3), a control ˆu∈Uwith X≜Xˆuminimizes

the functional Jif and only if Xsatisfies

X0=x, d ˙

Xt=1

κ(Xt−ξt)dt +dMtfor 0≤t≤T, ˙

XT= 0,(4.13)

for a suitable square integrable martingale (Mt)0≤t≤T.

2. In the constrained problem (4.4), a control ˆu∈UΞ

xwith X≜Xˆuminimizes

the functional Jif and only if Xsatisfies

X0=x, d ˙

Xt=1

κ(Xt−ξt)dt +dMtfor 0≤t < T, XT= ΞT,(4.14)

for a suitable square integrable martingale (Mt)0≤t<T .

In other words, the first order conditions in (4.13) and (4.14) are taking the form

of a coupled linear forward backward stochastic differential equation (FBSDE) for

the pair (X, u):

dXt=utdt,

dut=1

κ(Xt−ξt)dt +dMt,

with some square integrable martingale Msubject to

X0=xand ⎧

⎨

⎩

uT= 0 unconstrained case,

XT= ΞTconstrained case.

Proof. 1.) We start with the unconstrained problem (4.3). Since we are minimizing

the strictly convex functional u↦→ J(u)over U, a necessary and sufficient condition

for the optimality of ˆu∈Uwith corresponding Xˆu=x+∫·

0ˆusds is given by

⟨J′(ˆu), w⟩= 0 for all w∈U

101

4 Hedging with Temporary Price Impact

(cf., e.g., Ekeland and Témam (1999)). In view of Lemma 4.4.1 this means that

ˆu∈Uis optimal if and only if

E[∫T

ws(κˆus+∫T

(Xˆu

t−ξt)dt)ds]= 0 (4.15)

for all w∈U. We will now show that the first order condition in (4.15) is satisfied

(i.e., ˆu∈Uis optimal) if and only if Xˆusatisfies the dynamics in (4.13).

Necessity: Assume that ˆu∈Uwith Xˆu=x+∫·

0ˆusds minimizes J, i.e., condition

(4.15) is satisfied by ˆu. Then, by Fubini’s Theorem and optional projection, we

also get that

E[∫T

ws(κˆus+E[∫T

(Xˆu

t−ξt)dt⏐⏐⏐⏐

Fs])ds]= 0

for all w∈U. However, this is only possible if

ˆus=−1

κE[∫T

(Xˆu

t−ξt)dt⏐⏐⏐⏐

Fs]dP⊗ds-a.e. on Ω×[0, T].(4.16)

Hence, by defining the square integrable martingale

Ms≜E[∫T

(Xˆu

t−ξt)dt⏐⏐⏐⏐

Fs],0≤s≤T, (4.17)

we obtain the representation

ˆus=−1

κ(Ms−∫s

(Xˆu

t−ξt)dt)dP⊗ds-a.e. on Ω×[0, T],(4.18)

in other words, Xˆusatisfies the dynamics in (4.13). In particular, Xˆu

0=xand

Xˆu

T= ˆuT= 0 P-a.s.

Sufficiency: Assume now that ˆu∈Uwith corresponding Xˆusatisfies the dy-

namics in (4.13) with Xˆu

0=xand ˙

Xˆu

T= 0 P-a.s. Note that the unique strong

solution to this linear FBSDE in (4.13) is indeed given by (4.16) or, equivalently,

by (4.18). However, using this representation of ˆuand applying Fubini’s Theorem

yields

E[∫T

ws(κˆus+∫T

(Xˆu

t−ξt)dt)ds]=E[∫T

ws(MT−Ms)ds]

=E[∫T

wsE[MT−Ms|Fs]ds]=∫T

E[ws(E[MT|Fs]−Ms)]ds = 0

for all w∈U, since Mis a martingale. Consequently, the first order condition in

(4.15) is satisfied and ˆu∈Uis optimal.

102

4.4 Proofs

2.) Similar as above, a necessary and sufficient condition for the optimality of

ˆuΞ∈UΞ

xwith corresponding XˆuΞ=x+∫·

0ˆuΞ

sds satisfying XˆuΞ

T= ΞTP-a.s. for

the constrained problem (4.4) is given by

⟨J′(ˆuΞ), w⟩= 0 for all w∈U0

In contrast to the unconstrained case, observe now that we have an additional

constraint on w. Again, in view of Lemma 4.4.1, we get that ˆuΞ∈UΞ

xis optimal

if and only if

E[∫T

ws(κˆuΞ

s+∫T

(XˆuΞ

t−ξt)dt)ds]= 0 for all w∈U0

0.(4.19)

We will now show that the first order condition in (4.19) is fulfilled (i.e., ˆuΞ∈UΞ

is optimal) if and only if XˆuΞsatisfies the dynamics in (4.14).

Sufficiency: Assume that ˆuΞ∈UΞ

xwith corresponding XˆuΞsatisfies the dynam-

ics in (4.14) with XˆuΞ

0=xand XˆuΞ

T= ΞTP-a.s. That is, we have the representation

ˆuΞ

t= ˆuΞ

0+1

κ∫t

(XˆuΞ

s−ξs)ds +MtdP⊗dt-a.e. on Ω×[0, T)

for some square integrable martingale (Mt)0≤t<T . From ˆuΞ, ξ ∈L2(P⊗dt), it

follows that E[∫T

0M2

sds]<∞. Defining the square integrable martingale

NΞ

s≜E[∫T

(XˆuΞ

t−ξt)dt⏐⏐⏐⏐

Fs],0≤s≤T,

the above representation of ˆuΞyields

E[∫T

ws(κˆuΞ

s+∫T

(XˆuΞ

t−ξt)dt)ds]

=E[∫T

ws(κˆuΞ

0+NΞ

T+κMs)ds]

=E[(κˆuΞ

0+NΞ

T)∫T

wsds]+κE[∫T

wsMsds]

= 0 for all w∈U0

by virtue of Lemma 4.4.3 below. Consequently, the first order condition in (4.19)

is satisfied and ˆuΞ∈UΞ

xis optimal.

Necessity: As shown in the proof of Theorem 4.2.2 below (which does not use the

necessity assertion of the present Lemma), the optimal control ˆuΞin (4.9) satisfies

the dynamics in (4.14). Moreover, by strict concavity of the objective functional

in (4.2), the solution to problem (4.4) is unique. Therefore, the assertion is indeed

necessary.

103

4 Hedging with Temporary Price Impact

The following technical Lemma is needed in the proof of Lemma 4.4.2 for the

constrained problem (4.3).

Lemma 4.4.3. Let Mbe an adapted càdlàg process on [0, T)with E[∫T

0M2

sds]<

∞. Then,

E[∫T

wsMsds]= 0 for all w∈U0

0(4.20)

if and only if Mis a square integrable martingale on [0, T ).

Proof. First, assume that Mis a square integrable martingale on [0, T)with E[∫T

0M2

sds]<

∞. Consider a w∈U0

0such that w= 0 on Ω×[T−ε, T]for some ε > 0. Then,

by applying Fubini’s Theorem we have

E[∫T

wsMsds]=E[∫T−ε

wsE[MT−ε|Fs]ds]=E[MT−ε∫T

wsds]= 0.

Now, let w∈U0

0be arbitrary and consider an approximating sequence (w(n))n≥1⊂

0with w(n)= 0 on Ω×[T−εn, T]for some εn↓0such that w(n)→win

L2(Ω ×[0, T],P⊗dt)for n→ ∞. Then, by the Cauchy-Schwarz inequality we

obtain

lim

n→∞E[∫T

0|(w(n)

s−ws)Ms|ds]= 0.

Consequently,

E[∫T

wsMsds]= lim

n→∞E[∫T

w(n)

sMsds]= 0,

where the last identity follows from our initial consideration for ws with support

in [T−ε, T],ε > 0. Hence, the condition in (4.20) is satisfied.

Conversely, assume now that the condition in (4.20) is satisfied. We have to show

that Mis a square integrable martingale on [0, T). Let 0≤t<u<T,A∈Ft, be

arbitrary. For any ε > 0such that t+ε, u +ε < T we define

wε

s(ω)≜1A(ω)1

ε(1[t,t+ε](s)−1[u,u+ε](s))on Ω×[0, T].

Obviously, wis progressively measurable, in L2(P⊗ds)and satisfies ∫T

0wsds = 0

P-a.s. Hence, by assumption (4.20) we have

0 = E[∫T

wε

sMsds]=E[1A

ε∫t+ε

Msds]−E[1A

ε∫u+ε

Msds].

Passing to the limit ε↓0, we obtain by right-continuity of M,

0 = E[1A(Mt−Mu)] for all 0≤t < u < T.

Consequently, Mis a martingale on [0, T). By assumption, we have E[∫T

0M2

sds]<

∞which implies that Mis square integrable on [0, T).

104

4.4 Proofs

Now, we are ready to prove our main result by simple verification. We start with

Theorem 4.2.1 for the unconstrained problem (4.3).

Proof of Theorem 4.2.1.We divide the proof in two parts. First, we prove

optimality of the solution given in (4.6). Then, we compute the corresponding

minimal costs given in (4.8).

Optimality of (4.6):In order to show that our candidate in (4.6) is the optimal

solution for problem (4.3), we need to check the first order condition in Lemma

4.4.2 1.). For this, define the processes

Yt≜∫t

ξscosh(τκ(s))ds and ˜

Mt≜E[YT|Ft],0≤t≤T.

Since YT∈L2(P), we have that (˜

Mt)0≤t≤Tis a square integrable martingale. More-

over, note that Y, ˜

M∈L2(P⊗dt). Hence, the process ˆ

ξin Theorem 4.2.1 can be

written as

ξt=1

√κsinh(τκ(t)) (˜

Mt−Yt)dP⊗dt-a.e. on Ω×[0, T)(4.21)

with corresponding dynamics

dˆ

ξt=−coth(τκ(t))

√κ(ξt−ˆ

ξt)dt +1

√κsinh(τκ(t))d˜

Mton [0, T).(4.22)

Due to Lemma 4.4.5 b), we know that ˆ

ξ∈L2(P⊗dt). Now, the density of the

solution from (4.6) satisfies

dˆut=−1

κ(1 −tanh(τκ(t))2)(ˆ

ξt−ˆ

Xt)dt +1

√κtanh(τκ(t))(dˆ

ξt−dˆ

Xt)

κ((ˆ

Xt−ξt)dt +1

cosh(τκ(t))d˜

Mt)dP⊗dt-a.e. on Ω×[0, T],

that is, ˆusatisfies the BSDE-dynamics in (4.13). Obviously, it holds that ˆ

X0=x.

Solving equation (4.6) for ˆ

Xyields upon differentiation

ˆut=−1

√κ

sinh(τκ(t))

cosh(τκ(0))x

−1

κsinh(τκ(t))∫t

ξs

sinh(τκ(s))

cosh(τκ(s))2ds +1

Mt−Yt

cosh(τκ(t)) (4.23)

and we observe that limt↑Tˆut= 0 P-a.s., i.e., the terminal condition in (4.13) is

indeed satisfied. It remains to show that ˆu∈L2(P⊗dt). Since ˜

M, Y ∈L2(P⊗dt),

it suffices to observe that sinh(τκ(s))/cosh(τκ(s))2is bounded and therefore

E[∫T

0(∫t

ξs

sinh(τκ(s))

cosh(τκ(s))2ds)2]dt ≤const E[∫T

0(∫t

0|ˆ

ξs|ds)2

dt]

≤const T2

2∥ˆ

ξ∥2

L2(P⊗dt)<∞.

105

4 Hedging with Temporary Price Impact

Computation of minimal costs: To compute the minimal costs associated to the

optimal control ˆugiven in (4.8), note first that ˆu∈L2(P⊗dt)implies ˆ

X∈L2(P⊗dt)

and thus J(ˆu)<∞. For ease of presentation, we define

c(t)≜√κtanh(τκ(t)),0≤t≤T,

so that ˆut=c(t)(ˆ

ξt−ˆ

Xt)/κ. Hence, the minimal costs can be written as

∞> J(ˆu) = E[1

2∫T

(ˆ

Xs−ξs)2ds +1

2κ∫T

ˆu2

sds]

=lim

t↑T{1

2E[∫t

sds]−E[∫t

Xsξsds]+1

2E[∫t

ξ2

sds]

2κE[∫t

c(s)2ˆ

ξ2

sds]−1

κE[∫t

c(s)2ˆ

Xsˆ

ξsds]

2κE[∫t

c(s)2ˆ

sds]},(4.24)

due to monotone convergence. Observe that, using integration by parts and the

dynamics of ˆ

ξfrom (4.22), we have, for all t < T,

E[c(t)ˆ

t] =c(0)x2+2

κE[∫t

c(s)2ˆ

Xsˆ

ξsds]

−1

κE[∫t

c(s)2ˆ

sds]−E[∫t

sds]

as well as

E[c(t)ˆ

Xtˆ

ξt] =c(0)ˆ

ξ0x+1

κE[∫t

c(s)2ˆ

ξ2

sds]−E[∫t

Xsξsds]

and

E[c(t)ˆ

ξ2

t] =c(0)ˆ

ξ2

0+1

κE[∫t

c(s)2ˆ

ξ2

sds]−2E[∫t

ξsξsds]

+E[∫t

ξ2

sds]+E[∫t

c(s)d⟨ˆ

ξ⟩s].

Using these identities, we can write (4.24) as

∞> J(ˆu) = lim

t↑T{1

2c(0)(x−ˆ

ξ0)2+1

2E[∫t

(ˆ

ξs−ξs)2ds]

2E[∫t

c(s)d⟨ˆ

ξ⟩s]−1

2c(t)E[( ˆ

Xt−ˆ

ξt)2]}.(4.25)

To conclude our assertion for the minimal costs in (4.8), observe that

E[( ˆ

Xt−ˆ

ξt)2]≤2(E[ˆ

t] + E[ˆ

ξ2

t]),

and let us argue why

lim

t↑Tc(t)E[ˆ

t] = 0 and lim

t↑Tc(t)E[ˆ

ξ2

t] = 0.(4.26)

106

4.4 Proofs

By Jensen’s inequality, we have

E[ˆ

t]≤tE[∫t

ˆu2

sds]≤TE[∫T

ˆu2

sds]<∞.

Hence, due to limt↑Tc(t) = 0, the first convergence in (4.26) holds true. Concerning

the second convergence in (4.26), we use the representation in (4.21) for ˆ

ξto obtain,

again with Jensen’s inequality as well as the Cauchy-Schwarz inequality,

0≤c(t)E[ˆ

ξ2

t] = c(t)

κsinh(τκ(t))2E[( ˜

Mt−Yt)2]

≤c(t)

κsinh(τκ(t))2E[(YT−Yt)2]

=c(t)

κsinh(τκ(t))2E⎡

⎣(∫T

ξscosh(τκ(s))ds)2⎤

⎦

≤cosh(τκ(0))2

√κcosh(τκ(t))

sinh(τκ(t))(T−t)E[∫T

ξ2

sds]

≤cosh(τκ(0))2

cosh(τκ(t)) E[∫T

ξ2

sds]−→

t↑T0,

where for the last inequality we used that sinh(τ)≥τfor all τ≥0. In other words,

also the second convergence in (4.26) holds true. This finishes our proof of the

representation of the minimal costs in (4.8).

Next, we come to the proof of Theorem 4.2.2 concerning the constrained problem

(4.4).

Proof of Theorem 4.2.2.Again, we will proceed in two steps. First, we prove

optimality of the solution given in (4.9). Then, we compute the corresponding

minimal costs given in (4.11).

Optimality of (4.9):The verification of the optimality of ˆ

XΞ=x+∫·

0ˆuΞ

tdt in

Theorem 4.2.2 for the constrained problem (4.4) follows along the same lines as in

the unconstrained case. Again, we have to check the first order condition in Lemma

4.4.2 2.). For this, we define the processes

Yt≜1

√κ∫t

ξssinh(τκ(s))ds and ˜

MΞ

t≜E[YT+ ΞT|Ft]

for all 0≤t≤T. Since ZT,Ξ∈L2(P), we have that (˜

MΞ

t)0≤t≤Tis a square

integrable martingale. Moreover, note that Y, ˜

MΞ∈L2(P⊗dt). Hence, the process

ξΞin Theorem 4.2.2 can be written as

ξΞ

t=1

cosh(τκ(t)) (˜

MΞ

t−Yt)dP⊗dt-a.e. on Ω×[0, T](4.27)

107

4 Hedging with Temporary Price Impact

with corresponding dynamics

dˆ

ξΞ

t=−tanh(τκ(t))

√κ(ξt−ˆ

ξΞ

t)dt +1

cosh(τκ(t))d˜

MΞ

ton [0, T].(4.28)

In particular, we observe that ˆ

ξΞ∈L2(P⊗dt). Similar to the unconstrained case

above, one easily checks that

dˆuΞ

t=1

κ(ˆ

XΞ

t−ξt)dt +1

√κ

sinh(τκ(t))d˜

MΞ

tdP⊗dt-a.e. on Ω×[0, T),

that is, ˆuΞsatisfies the dynamics in (4.14). Obviously, it holds that ˆ

XΞ

0=x.

Next, we have to check the terminal condition in (4.14), that is, limt↑Tˆ

XΞ

t= ΞT

P-a.s. In order to show this, first note that we can consider a càdlàg version of

(ˆ

ξΞ

t)0≤t≤Tdue to its representation in (4.27). Hence, since ΞTis FT−-measurable

by assumption (4.5) we obtain the P-a.s. limit

lim

t↑T

ξΞ

t=E[ΞT|FT−] = ΞT

in (4.27). In other words, for every ε > 0there exists a random time Υε∈[0, T)

such that P-a.s.

ΞT−ε≤ˆ

ξΞ

t≤ΞT+εfor all t∈[Υε, T].

For limt↑Tˆ

XΞ

t= ΞTP-a.s., it clearly suffices to show that for any ε > 0it holds

that

lim sup

t↑T

XΞ

t≤ΞT+εand lim inf

t↑T

XΞ

t≥ΞT−εP-a.s.

Define Xε

t≜ΞT+ε−ˆ

XΞ

tso that ˆ

ξΞ

t−ˆ

XΞ

t≤Xε

tP-a.s. for t∈[Υε, T). This yields

dXε

t=−dˆ

XΞ

t=−1

√κcoth(τκ(t))(ˆ

ξΞ

t−ˆ

XΞ

t)dt

≥− 1

√κcoth(τκ(t))Xε

tdt.

Moreover, note that for all ω∈Ωthe linear ODE on [Υε(ω), T)given by

dZt=−1

√κcoth(τκ(t))Ztdt, ZΥε(ω)=Xε

Υε(ω)(ω),

admits the solution

Zt=Xε

Υεexp(−1

√κ∫t

Υε

coth(τκ(s))ds)=Xε

Υε

sinh(τκ(t))

sinh(τκ(Υε)), t < T,

with limt↑TZt= 0. By the comparison principle for ODEs, we get P-a.s. Xε

t≥Zt

for all t∈[Υε, T). Hence,

lim inf

t↑TXε

t≥lim

t↑TZt= 0 P-a.s.,

108

4.4 Proofs

that is, lim supt↑Tˆ

XΞ

t≤ΞT+εP-a.s. Similarly, define ˜

Xε

t≜ΞT−ε−ˆ

XΞ

tand

observe as above that P-a.s. on [Υε, T )we have

d˜

Xε

t≤ − 1

√κcoth(τκ(t)) ˜

Xε

tdt.

Again, as above by comparison principle we obtain

lim sup

t↑T

Xε

t≤0P-a.s.,

i.e., lim inft↑Tˆ

XΞ

t≥ΞT−εP-a.s. as remained to be shown for (4.14).

Finally, we have to argue that ˆuΞ∈L2(P⊗dt). For this, we may assume without

loss of generality that x= 0. Moreover, let us denote ˆuΞ,ξ ≜ˆuΞ,ˆ

XΞ,ξ ≜ˆ

XΞand

ξΞ,ξ ≜ˆ

ξΞto emphasize also the dependence on the given target process ξ. With

this notation it holds that

ˆuΞ,ξ = ˆuΞ,0+ ˆu0,ξ.

Hence, we have to show that ˆuΞ,0∈L2(P⊗dt)and ˆu0,ξ ∈L2(P⊗dt).

Concerning ˆuΞ,0, observe that, using ˆ

ξΞ,0

t= Ξt/cosh(τκ(t)) with Ξt≜E[ΞT|Ft],

0≤t≤T, as well as the explicit solution ˆ

XΞ,0

tfor the ODE in (4.9), we obtain

ˆuΞ,0

t=coth(τκ(t))

√κ(ˆ

ξΞ,0

t−ˆ

XΞ,0

=coth(τκ(t))

√κ(e−∫t

coth(τκ(u))

√κdu ˆ

ξΞ,0

e−∫t

coth(τκ(u))

√κdu ∫t

e∫s

coth(τκ(u))

√κdudˆ

ξΞ,0

=cosh(τκ(t))

√κsinh(τκ(0)) ˆ

ξΞ,0

0+cosh(τκ(t))

κ∫t

Ξs

cosh(τκ(s))2ds

+cosh(τκ(t))

√κ∫t

sinh(2τκ(s))dΞs,(4.29)

where we used integration by parts in the second line. Obviously, the first two

terms in (4.29) belong to L2(P⊗dt). The third term is in L2(P⊗dt)since, using

Fubini’s Theorem as well as sinh(τ)≥τfor all τ≥0, we get

E[∫T

0(∫t

2dΞs

sinh(2τκ(s)))2

dt]=E[∫T

0∫t

0(2

sinh(2τκ(s)))2

d⟨Ξ⟩sdt]

=E[∫T

(T−s)(2

sinh(2τκ(s)))2

d⟨Ξ⟩s]≤E[∫T

T−sd⟨Ξ⟩s]

=κ∫T

dE[Ξ2

T−s<∞

by assumption (4.5).

109

4 Hedging with Temporary Price Impact

Concerning ˆu0,ξ, we use the explicit expressions for ˆ

ξ0,ξ

tand ˆ

X0,ξ

tto obtain in

(4.9) that

ˆu0,ξ

t=coth(τκ(t))

√κ(ˆ

ξ0,ξ

t−ˆ

X0,ξ

=cosh(τκ(t)) −1

√κsinh(τκ(t)) E[∫T

ξuKΞ(t, u)du⏐⏐⏐Ft]

−cosh(τκ(t))

κ∫t

cosh(τκ(s)) −1

sinh(τκ(s))2E[∫T

ξuKΞ(s, u)du⏐⏐⏐Fs]ds. (4.30)

Note that all the ratios in (4.30) involving the functions cosh(·)and sinh(·)are

actually bounded on [0, T]. Moreover, we have by Lemma 4.4.5 c) below that

E[∫T

ξuKΞ(t, u)du⏐⏐⏐Ft]∈L2(P⊗dt),

as well as, using Jensen’s inequality,

E⎡

⎣∫T

0(∫t

E[∫T

ξuKΞ(s, u)du⏐⏐⏐Fs]ds)2

dt⎤

⎦

≤T2

2E⎡

⎣∫T

0(E[∫T

ξuKΞ(s, u)du⏐⏐⏐Fs])2

ds⎤

⎦<∞.

Together, this shows ˆuΞ∈L2(P⊗dt)as desired.

Computation of minimal costs: Now, we compute the minimal costs associated

to the optimal control ˆuΞgiven in (4.11). We will follow along the same lines as

in the unconstrained case above. First of all, note that ˆuΞ∈L2(P⊗dt)implies

XΞ∈L2(P⊗dt)and hence J(ˆu)<∞. For ease of presentation, we define

c(t)≜√κcoth(τκ(t)),0≤t < T,

i.e., ˆuΞ

t=c(t)(ˆ

ξΞ

t−ˆ

XΞ

t)/κ. Analogously to the unconstrained case above, we can

write J(ˆuΞ)as

∞> J(ˆuΞ) = lim

t↑T{1

2c(0)(x−ˆ

ξΞ

0)2+1

2E[∫t

(ˆ

ξΞ

s−ξs)2ds]

2E[∫t

c(s)d⟨ˆ

ξΞ⟩s]−1

2c(t)E[( ˆ

XΞ

t−ˆ

ξΞ

t)2]}.(4.31)

To conclude our assertion for the minimal costs in (4.11), observe that

E[( ˆ

XΞ

t−ˆ

ξΞ

t)2]≤2(E[( ˆ

XΞ

t−Ξt)2] + E[(Ξt−ˆ

ξΞ

t)2]),

where Ξt≜E[ΞT|Ft],0≤t≤T, and let us argue why

lim

t↑Tc(t)E[( ˆ

XΞ

t−Ξt)2] = 0 and lim

t↑Tc(t)E[(Ξt−ˆ

ξΞ

t)2] = 0.(4.32)

110

4.4 Proofs

Concerning the first convergence in (4.32), Jensen’s inequality, monotonicity of

the function cosh(·)as well as the estimate sinh(τ)≥τfor all τ≥0yield

c(t)E[( ˆ

XΞ

t−Ξt)2]≤c(t)E[( ˆ

XΞ

t−ˆ

XΞ

T)2]

≤κcosh(τκ(0))

T−tE⎡

⎣(∫T

ˆuΞ

sds)2⎤

⎦

≤κcosh(τκ(0))E[∫T

(ˆuΞ

s)2ds]−→

t↑T0,(4.33)

since ΞT=ˆ

XΞ

Tand ˆuΞ∈L2(P⊗dt).

Concerning the second convergence in (4.32), we insert the definition for ˆ

ξΞto

obtain that

c(t)E[(Ξt−ˆ

ξΞ

t)2]

=c(t)E[(cosh(τκ(t)) −1

cosh(τκ(t)) Ξt

−cosh(τκ(t)) −1

cosh(τκ(t)) E[∫T

ξuKΞ(t, u)du⏐⏐⏐Ft])2⎤

⎦

≤2c(t)(cosh(τκ(t)) −1

cosh(τκ(t)) )2

E[Ξ2

+ 2c(t)(cosh(τκ(t)) −1

cosh(τκ(t)) )2

E[∫T

ξ2

uKΞ(t, u)du]

≤2√κ

cosh(τκ(t))

(cosh(τκ(t)) −1)2

sinh(τκ(t)) E[Ξ2

+2sinh(τκ(0))

cosh(τκ(t))

cosh(τκ(t)) −1

sinh(τκ(t)) E[∫T

ξ2

udu]−→

t↑T0,

since ΞT∈L2(P),ξ∈L2(P⊗dt)and limt↑T(cosh(τκ(t)) −1)/sinh(τκ(t)) = 0.

Consequently, also the second convergence in (4.32) holds true. This finishes our

proof of the representation of the minimal costs in (4.11).

The next Lemma shows that the set UΞ

xis not empty under the assumption

(4.5).

Lemma 4.4.4. For ΞT∈L2(P,FT)we have that UΞ

x=∅if and only if condition

(4.5)holds, i.e., if and only if ∫T

dE[Ξ2

T−t<∞with Ξt≜E[ΞT|Ft]for all 0≤t≤T.

Proof. Let ΞT∈L2(P,FT). We first prove necessity. Assume there exists u∈UΞ

i.e., u∈L2(P⊗dt)such that

T=x+∫T

usds = ΞT.

111

4 Hedging with Temporary Price Impact

Then, applying Fubini’s Theorem, we obtain

∫T

dE[Ξ2

T−t=1

T(E[Ξ2

T]−E[Ξ2

0]) + ∫T

E[Ξ2

T−Ξ2

s]d(1

T−s).

Moreover, E[Ξ2

T−Ξ2

s] = E[(ΞT−Ξs)2]≤E[(Xu

T−Xu

s)2]due to the L2-projection

property of conditional expectations. Hence, we get

∫T

dE[Ξ2

T−t≤1

T(E[Ξ2

T]−E[Ξ2

0]) + ∫T

E⎡

⎣(∫T

urdr)2⎤

⎦d(1

T−s)

T(E[Ξ2

T]−E[Ξ2

0]) + E⎡

⎣∫T

0(1

T−s∫T

urdr)2

ds⎤

⎦<∞

by ΞT∈L2(P)and Lemma 4.4.5 a).

For sufficiency, simply consider the optimizer ˆuΞfrom Theorem 4.2.2 which we

proved to be in UΞ

xunder the condition (4.5).

The final Lemma collects estimates concerning the L2(P⊗dt)-norm which are

needed several times in the proofs above.

Lemma 4.4.5. Let (ζt)0≤t≤T∈L2(P⊗dt)be progressively measurable. Moreover,

let K(t, u),KΞ(t, u),0≤t≤u < T, denote the kernels from Theorems 4.2.1 and

4.2.2, respectively.

a) For ¯

ζt≜1

T−t∫T

tζsds,t < T, we have

∥¯

ζ∥L2(P⊗dt)≤2∥ζ∥L2(P⊗dt).

b) For ζK

t≜E[∫T

tζuK(t, u)du|Ft],t < T, we have

∥ζK∥L2(P⊗dt)≤c∥ζ∥L2(P⊗dt)

for some constant c > 0.

c) For ζKΞ

t≜E[∫T

tζuKΞ(t, u)du|Ft],t < T, we have

∥ζKΞ∥L2(P⊗dt)≤c∥ζ∥L2(P⊗dt)

for some constant c > 0.

112

4.4 Proofs

Proof. a) By Fubini’s Theorem and the Cauchy-Schwarz inequality, we have

∥¯

ζ∥2

L2(P⊗dt)=E[∫T

0∫T

ζrζs∫r∧s

0(1

T−t)2

dtdrds]

=E[∫T

0∫T

ζrζs

T−r∧sdrds]−1

TE⎡

⎣(∫T

ζsds)2⎤

⎦

≤E[2∫T

ζr∫r

ζs

T−sdsdr]

= 2E[∫T

ζs(1

T−s∫T

ζrdr)ds]

≤2∥ζ∥L2(P⊗dt)∥¯

ζ∥L2(P⊗dt)

and hence the assertion.

b) First, assume that (ζt)0≤t≤Tis deterministic, and so ζK

t=∫T

tζuK(t, u)du.

By similar computations as in a) we obtain

∥ζK∥2

L2(dt)=∫T

0∫T

ζrζs∫r∧s

K(t, r)K(t, s)dtdrds

≤∫T

0∫T

ζrζs

√κcosh(τκ(r)) cosh(τκ(s)) coth(τκ(r∧s))drds

= 2∫T

ζr

cosh(τκ(r))

√κ∫r

ζscosh(τκ(s)) coth(τκ(s))dsdr

= 2∫T

ζscosh(τκ(s))2ζK

sds

≤2cosh(τκ(0))2∥ζ∥L2(dt)∥ζK∥L2(dt),

i.e., ∥ζK∥L2(dt)≤c∥ζ∥L2(dt)for some constant c > 0. Now, for general (ζt)0≤t≤T∈

L2(P⊗dt)progressively measurable, we get with Fubini’s Theorem

E[∫T

(ζK

t)2dt]=∫T

0∫T

t∫T

E[E[ζr|Ft]E[ζs|Ft]]K(t, r)K(t, s)drdsdt.

Again, application of Cauchy-Schwarz’s and Jensen’s inequalities yields

E[E[ζr|Ft]E[ζs|Ft]] ≤ ∥ζr∥L2(P)∥ζs∥L2(P), t ≤r, s ≤T.

Consequently,

∥ζK∥2

L2(P⊗dt)≤∫T

0∫T

t∫T

t∥ζr∥L2(P)∥ζs∥L2(P)K(t, r)K(t, s)drdsdt

=∫T

0(∫T

t∥ζr∥L2(P)K(t, r)dr)2

dt.

113

4 Hedging with Temporary Price Impact

Now, put ˜

ζt≜∥ζt∥L2(P)and apply the estimate already proved for deterministic

functions to conclude

∥ζK∥2

L2(P⊗dt)=∫T

0(∫T

ζrK(t, r)dr)2

≤c∫T

0|˜

ζt|2dt =c∫T

E[ζ2

t]dt =c∥ζ∥2

L2(P⊗dt).

c) Jensen’s inequality and Fubini’s Theorem give

∥ζKΞ∥2

L2(P⊗dt)=E[∫T

(ζKΞ

t)2dt]≤∫T

0∫T

E[ζ2

u]KΞ(t, u)dudt

=∫T

E[ζ2

u]∫u

KΞ(t, u)dtdu.

Now, using cosh(τ)−1≥τ2/2for all τ≥0, we get

0≤∫u

KΞ(t, u)dt =∫u

sinh(τκ(u))

√κ(cosh(τκ(t)) −1)dt

≤sinh(τκ(u))

√κ∫u

2κ

(T−t)2dt ≤2√κsinh(τκ(u))

T−u−→

u↑T1.

Thus, the above integral over KΞis bounded uniformly in 0≤u≤Tby some

constant c > 0, and so

∥ζKΞ∥2

L2(P⊗dt)≤c∫T

E[ζ2

u]du =c∥ζ∥2

L2(P⊗dt)

yielding the assertion in c).

114

5 General Stochastic Linear Quadratic Control for

Hedging with Temporary Price Impact

The optimal tracking problem studied in the preceding Chapter 4in the context

of hedging a European contingent claim in the presence of temporary price impact

as proposed by Almgren and Chriss (2001) is a prototype of a linear quadratic

stochastic optimal control problem (stochastic LQ problem in short); cf., e.g., the

book by Yong and Zhou (1999), Chapter 6, for an overview. In the present chapter,

we want to extend the results from Chapter 4to the more general case where we

allow for stochastic coefficients in the objective functional formulated in (4.2) above.

5.1 Problem setup

Let us fix a finite deterministic time horizon T > 0and a filtered probability

space (Ω,F,(Ft)0≤t≤T,P)satisfying the usual conditions of right continuity and

completeness. We let (κt)0≤t≤Tand (νt)0≤t≤Tdenote two progressively measurable,

strictly positive processes such that

∫T

0(νt+1

κt)dt < ∞P-a.s. (5.1)

Moreover, we are given a predictable target process (ξt)0≤t≤Tsatisfying

E[∫T

0|ξt|νtdt]<∞and ∫T

ξ2

tνtdt < ∞P-a.s. (5.2)

as well as a random terminal target position ΞT∈L0(P,FT−). Similar to the

problem formulation in Chapter 4, Section 4.1, we seek to minimize a cost criterium

of the following form: For a given x∈R, find a progressively measurable control

u∈L1([0, T], ds)P-a.s. with controlled state process

t=x+∫t

usdt (0 ≤t≤T)(5.3)

which minimizes the objective functional

E[∫T

(Xu

t−ξt)2νtdt +∫T

κtu2

tdt +η(Xu

T−ΞT)2](5.4)

where η∈FTdenotes a nonnegative random variable. The interpretation of the

LQ problem in (5.4) is as before in Chapter 4(recall also the motivation from Chap-

ter 3and the heuristically derived benchmark optimization problem in (3.9)): The

115

5 General Linear Quadratic Control for Hedging with Temporary Impact

state process (Xu

t)0≤t≤Tdenotes an agent’s position in some risky asset that she

trades at a turnover rate (ut)0≤t≤T. She wants her position to be as close as possible

to a given target strategy (ξt)0≤t≤Tbut simultaneously seeks to minimize the in-

duced quadratic transaction costs which are levied on her transactions due to, e.g.,

stochastic temporary price impact as measured by (κt)0≤t≤T. The weight process

(νt)0≤t≤Tcaptures stochastic volatility, that is, the risk of her open trading position

due to random market fluctuations. In addition, the third term in (5.4) implements

a penalization on the quadratic deviation of the agent’s terminal position Xu

Tfrom

a final target position ΞTwith random penalization parameter η∈FTsatisfying

P[0 ≤η≤+∞] = 1.(5.5)

In other words, similar to the (almost sure) constrained stochastic LQ problem

formulated in (4.4) in Chapter 4, we are interested in studying the optimization

problem in (5.4) which additionally incorporates a possibly singular stochastic ter-

minal state constraint. Specifically, on the event {η= +∞}, it is natural to expect

that the “blow up” of ηimposes a stochastic terminal state constraint of the form

T= ΞTa.e. on the set {η= +∞} (5.6)

on all controlled processes Xuthat produce a finite value in (5.4). Mathematically,

as we will discuss in Section 5.2 below, it is less obvious how to tackle this delicate

singularity and how to compute the optimal control as well as the optimal value.

In particular, it is not clear a priori whether a controlled process Xuwhich respects

the stochastic constraint in (5.6) and matches the random position ΞTon the event

{η= +∞} at terminal time Tdoes likewise entail finite expected quadratic costs

in (5.4). Put differently, the problem might not admit an optimal finite solution

at all. This has to be precluded via identifying appropriate conditions as, e.g., the

condition in (4.5) in Section 4.1 above on the terminal position ΞT.

In fact, having at hand the results from Chapter 4in the case of constant coef-

ficients νt≡ν∈R+,κt≡κ∈R+and η∈ {0,+∞}, we will formulate and solve a

suitable variant of the above LQ problem in (5.4) with singular stochastic terminal

state constraint in the sense of (5.6) in Sections 5.3 and 5.4 below. Recall that

Theorem 4.2.2 revealed that the optimal solution and the corresponding optimal

value to the constrained LQ problem in (4.4) can be characterized in a particularly

enlightening manner by the optimal signal process ˆ

ξΞdefined in (4.10). In the

present more general setting, given the solvability of a singular backward stochas-

tic Riccati differential equation (BSRDE) which we will introduce shortly, it turns

out that the same key role is played by a generalized version of this optimal signal

process. Specifically, this optimal signal process will provide the main tool not only

in solving but also in tackling the adressed LQ problem in (5.4) with its delicate

stochastic terminal state constraint and to resolve the technical difficulties it en-

tails. As we will see below it also allows similarly to the representation in (4.10) for

116

5.1 Problem setup

an intuitively appealing interpretation of the obtained optimal control and makes

transparent the associated optimal costs by generalizing the expressions obtained

in (4.9) and (4.11), respectively. As a byproduct, we identify necessary and suffi-

cient conditions under which our variant of the LQ problem in (5.4) with singular

stochastic terminal state constraint in the sense of (5.6) actually admits a finite

optimal value. These conditions can be considered as generalizing the condition

in (4.5) above on the terminal position ΞT(recall also Lemma 4.4.4).

Remark 5.1.1.1. Let us mention that the mild integrability conditions in (5.1)

and (5.2) ensure that all the processes to be introduced shortly are well defined

along with our stochastic LQ problem introduced in Section 5.3.

2. From a Mathematical Finance point of view, the terminal state constrained

in (5.6) can be interpreted as follows: In case of a possible but not necessarily

almost sure occurrence of specific market conditions, encoded by the event

set {η= +∞}, the agent may require to drive her risky asset position Xu

imperatively towards a predetermined random value ΞTat maturity T, e.g.,

to respect specific requirements of contractual or regulatory nature concerning

her risky asset position. Otherwise, a penalization depending on the deviation

of Xu

Tfrom the target position ΞTis implemented.

3. Stochastic control problems, referred to as optimal liquidation problems in

the literature, with almost sure singular (i.e., η= +∞almost surely) and

deterministic terminal state constraint (targeting the terminal position ΞT=

0), where the cost functional is allowed to be quadratic in the state process

Xuand the control u(that is, ξ≡0in (5.4)) have already been studied in,

e.g., Schied (2013), Ankirchner et al. (2014) and, in a more general BSPDE

framework, in Graewe et al. (2015); allowing the penalization parameter ηto

take the value infinity with positive probability has been investigated in Kruse

and Popier (2016a). Ankirchner and Kruse (2015), still within this context of

optimal liquidation, allow the objective functional to be additionally linear in

the control u. They also incorporate a specific non-zero stochastic terminal

state constraint where the random target position ΞTis gradually revealed up

to terminal time T. A general class of stochastic control problems including

LQ problems with terminal states being constrained to a convex set were

studied by Ji and Zhou (2006). However, to the best of our knowledge,

stochastic linear quadratic control problems with ξ= 0 and possibly singular,

general stochastic terminal state constraint ΞT= 0 as considered in the

present chapter have not yet been investigated.

4. Similar general optimal tracking problems with stochastic coefficients as the

one in (5.4) have been studied in the literature which we already mentioned in

Chapter 4, Remark 4.1.2 above: Recall, e.g., Rogers and Singh (2010), Nau-

jokat and Westray (2011), Almgren and Li (2016), Frei and Westray (2013),

117

5 General Linear Quadratic Control for Hedging with Temporary Impact

Cartea and Jaimungal (2016), Cai et al. (2015) and, from an asymptotic anal-

ysis perspective, in Chan and Sircar (2016). Note, however, that the above

cited papers may neither allow for an arbitrary predictable target strategy ξ

nor for stochastic price impact κand stochastic volatility ν. In particular,

none of them consider a possibly singular stochastic terminal state constraint

in the sense of (5.6) above with general random target position ΞT.

5.2 Connection between stochastic LQ problems and BS(R)DEs

In case where the penalization parameter ηin the objective functional formulated

in (5.4) is bounded, it is well known in the literature that the optimal control to

the stochastic LQ problem as well as its optimal value is fully characterized by

two coupled backward stochastic differential equations (BSDEs); cf., e.g., Bismut

(1976,1978): A backward stochastic Riccati differential equation (BSRDE) of the

form

dct=(c2

κt−νt)dt −dNton [0, T]with cT=η(5.7)

and, due to the linear component in the objective functional in (5.4), a linear BSDE

of the form

dbt=(ct

κt

bt−νtξt)dt +dMton [0, T]with bT=ηΞT,(5.8)

where (Nt)0≤t≤T,(Mt)0≤t≤Tdenote some càdlàg local martingales on the under-

lying filtered probability space (Ω,F,(Ft)0≤t≤T,P); cf., e.g., Kohlmann and Tang

(2002), Section 5.1.

However, note that in our case with P[0 ≤η≤+∞] = 1 the involved BS(R)DEs

in (5.7) and (5.8) will both now exhibit with positive probability a singularity at

final time in this case. The solvability of the possibly singular BSRDE has been

recently studied in Kruse and Popier (2016a) and Popier (2016) (cf. also Graewe

et al. (2015) in the case η= +∞P-a.s.) and we will assume the solution process to

be given. In contrast, the singularity in the terminal condition of the linear BSDE

in (5.8) is rather unpleasant because it also involves the desired target position ΞT.

As a consequence, one needs to find a suitable substitute for the linear BSDE.

The following standing assumption summarizes what we need to know about the

singular BSRDE in (5.7) for our purposes:

Assumption 5.2.1. There exists a unique (Ft)0≤t<T -adapted, càdlàg semimartin-

gale (ct)0≤t<T with BSRDE dynamics

dct=(c2

κt−νt)dt −dNton [0, T)(5.9)

for some càdlàg local martingale (Nt)0≤t<T and (possibly) singular terminal con-

dition

lim

t↑Tct=ηP-a.s. (5.10)

118

5.2 Connection between stochastic LQ problems and BS(R)DEs

The pair (c, N)satisfies

E[sup

s∈[0,t]|cs|2+ [N]t]<∞for all 0≤t < T. (5.11)

Moreover, it holds that

ct>0P-a.s. for all 0≤t < T (5.12)

and

∫[0,T )

d[c]t

t−

<∞on the set {η= +∞},(5.13)

where [c]denotes the quadratic variation process of the càdlàg semimartingale c

(cf., e.g., Protter (2004), Chapter II.6, for the quadratic variation process of càdlàg

semimartingales).

Remark 5.2.2.

1. Note that the dynamics in (5.9) have to be understood in the sense that for

all 0≤s≤t < T the pair (c, N)satisfies

cs=ct−∫t

s(c2

κu−νu)du +∫t

dNuP-a.s.

In particular, the dynamics in (5.9) are only required to hold on [0, T −ε]for

every ε > 0, that is, strictly before T.

2. Let us mention that in the special non-singular case where the random vari-

able ηis simply bounded P-a.s., existence and uniqueness results (within a

Brownian framework and for bounded processes (νt)0≤t≤Tand (κt)0≤t≤T) to

the above BSRDE in (5.9) with terminal condition cT=ηP-a.s. can be

found, e.g., in Kohlmann and Tang (2002). The corresponding solution pair

(c, N)satisfies property (5.11). Sufficient conditions under which the solution

process (ct)0≤t≤Tis strictly positive, i.e., condition (5.12) holds true, are also

provided therein.

3. In the singular case η= +∞P-a.s. and again within a Brownian framework,

existence and uniqueness results (under suitable integrability conditions on

the processes (νt)0≤t≤Tand (κt)0≤t≤T) to the above BSRDE in (5.9) with

singular terminal condition limt↑Tct= +∞P-a.s. are provided in Ankirchner

et al. (2014) (cf. also Graewe et al. (2015)). Therein, the solution pair (c, N)

satisfies likewise conditions (5.11) and (5.12).

4. In the present partial singular setup where the random variable ηis allowed

to take the value +∞with positive probability but not necessarily P-a.s.,

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5 General Linear Quadratic Control for Hedging with Temporary Impact

Kruse and Popier (2016a) provide sufficient conditions (including suitable in-

tegrability conditions on (κt)0≤t≤Tand (νt)0≤t≤T) for the existence of a min-

imal weak supersolution to the above BSRDE in (5.9) with slightly weaker

(possibly) singular terminal condition lim inft↑Tct≥ηP-a.s. (cf. also Ap-

pendix 5.6 below). Sufficient conditions on the random variable ηand the

process (κt)0≤t≤Tunder which the solution process (ct)0≤t<T possesses a left

limit as t↑Twhich is equal to η(as required in (5.10)) are discussed in Popier

(2016). The solution pair (c, N)provided in Kruse and Popier (2016a) satis-

fies the integrability condition in (5.11) and the solution process (ct)0≤t<T is

shown to be nonnegative. In Appendix 5.6, we will provide within the setup

of Kruse and Popier (2016a) a simple lower bound on the solution process

which gives sufficient conditions under which strict positivity of the process

con [0, T), i.e., property (5.12), is guaranteed (cf. Lemma 5.6.1 below).

5. Concerning the integrability condition in (5.13) on the “blow up” set {η=

+∞}, it is implicitly shown in Popier (2006) in a Brownian framework and

in the special case of constant coefficients ν≡0and κ≡1that this condi-

tion is indeed satisfied by the solution process (ct)0≤t<T of the corresponding

BSRDE (5.9) (cf. Theorem 2 and Proposition 3 in Popier (2006)). The other

above cited papers Kruse and Popier (2016a), Popier (2016) and Ankirchner

et al. (2014) do not further investigate this property, though. Since the re-

quired integrability in (5.13) is needed in the proof of Lemma 5.2.3 below

whose result feeds crucially into our solution presented in Section 5.4, we will

briefly discuss exemplarily in Appendix 5.6 within the framework of Kruse

and Popier (2016a) sufficient conditions on (κt)0≤t≤T,(νt)0≤t≤Tand ηunder

which property (5.13) does hold true.

Inspired by the results from Chapter 4, it turns out that it will be suitable to

introduce a generalized version of the optimal signal process from (4.10) above

which will serve as a proper substitute of the classical linear BSDE in (5.8) in

the unconstrained case with bounded penalization parameter η. This process will

be our main tool in tackling and solving our variant of the LQ problem in (5.4)

with singular stochastic terminal state constraint (5.6). In order to introduce this

process, let us first define the adjoint process

Lt≜cte−∫t

κudu (0 ≤t < T).(5.14)

We refer to the discussion after Definition 5.2.4 below for an explanation of this

terminology.

Lemma 5.2.3. The adjoint process (Lt)0≤t<T is a strictly positive càdlàg super-

martingale. In particular,

LT≜lim

t↑TLt≥0exists P-a.s. (5.15)

120

5.2 Connection between stochastic LQ problems and BS(R)DEs

and we have that (Lt)0≤t≤Tis a supermartingale on [0, T ]. Moreover, we have

{LT= 0}={η= 0}up to P-null sets.

Proof. Since ct>0P-a.s. for all 0≤t < T by condition (5.12), it is immediate

from (5.14) that also Lt>0P-a.s. for all 0≤t < T. Integration by parts and

using the dynamics of cin (5.9) yields that Lsatisfies the dynamics

L0=c0, dLt=Lt−(−νt

ct−

dt −1

ct−

dNt)on [0, T).(5.16)

Since Nis a càdlàg local martingale on [0, T), we obtain from (5.16) that the

process Lis a càdlàg supermartingale on [0, T). Hence, it follows by the (super-

)martingale convergence theorem (see, e.g., Karatzas and Shreve (1991), Chapter

1.3, Problem 3.16) that the limit LT≜limt↑TLtexists P-a.s. and extends the

process Lto a càdlàg supermartingale on all of [0, T]. Moreover, appealing to the

definiton of Lin (5.14) and the convergence of the process ctowards ηas t↑Tin

condition (5.10), we obviously have LT= 0 on the set {η= 0}as well as LT>0

on the set {0< η < ∞}. Concerning the “blow up” set {η= +∞}, observe that

we may write

Lt=c0eXt−1

2[X]c

t∏

s≤t

(1 + ∆Xs)e−∆Xs(0 ≤t < T)(5.17)

where Xt≜−∫t

0νs

cs−ds −∫t

cs−dNs(cf., e.g., Protter (2004), Theorem II.37). Con-

dition (5.12) guarantees ∆Xs>−1for all 0≤s < T. Moreover, applying Taylor’s

formula, it holds for all 0≤t < T that

∑

s≤t⏐⏐⏐log ((1 + ∆Xs)e−∆Xs)⏐⏐⏐≤1

2∫[0,T )

s−

d[c]s<+∞(5.18)

on the set {η= +∞} by virtue of condition (5.13). This implies that the product of

the jumps in (5.17) will converge to a strictly positive limit as t↑Ton {η= +∞}.

Concerning the limiting behaviour of the exponential exp(Xt−1

2[X]c

t)in (5.17) for

t↑T, observe that once more condition (5.13) prevents the limiting value from

becoming 0 on {η= +∞}. Indeed, the local martingale ∫t

0dNs/cs−cannot explode

as t↑Tfor those paths along which its quadratic variation ∫t

0d[c]s/c2

s−remains

bounded on [0, T)(cf., e.g., Protter (2004), Chapter V.2, for more details).

Concerning the predetermined terminal target position ΞT, we will henceforth

additionally assume that

ΞTLT∈L1(FT−,P).(5.19)

Now, we are in a position to introduce the key object of our approach:

Definition 5.2.4. For (ξt)0≤t≤Tand ΞTsatisfying (5.2) and (5.19), respectively,

we define the optimal signal process as the càdlàg semimartingale on [0, T)given

ξt≜1

E[ΞTLT+∫T

ξre−∫r

κuduνrdr ⏐⏐⏐⏐

Ft](0 ≤t < T).(5.20)

121

5 General Linear Quadratic Control for Hedging with Temporary Impact

Observe that ˆ

ξcan also be understood as the solution process to a linear BSDE

with adjoint process L(cf., e.g., the book by Pham (2009), Section 6.2.2, on the

explicit solution to linear BSDEs in a Brownian framework). But the process ˆ

may not possess a well defined terminal value in T. Indeed, by the definition in

(5.20) we immediately observe that

lim

t↑T(ˆ

ξtLt) = ΞTLTP-a.s.

by the martingale convergence theorem. Hence, due to the convergence of the

process Lin Lemma 5.2.3, we can deduce that

∃lim

t↑T

ξt= ΞTon the set {0< η ≤+∞}.(5.21)

In other words, the optimal signal process converges to the predetermined target

position ΞTas t↑Toutside of the set {η= 0}. By contrast, on the set {η= 0},

which is not necessarily a P-null set, we have LT= 0 by virtue of Lemma 5.2.3

and hence a limit of the process ˆ

ξtas t↑Tdoes not have to exist. As our analysis

shows, though, the fact that the process ˆ

ξmay not possess a well defined terminal

value in Tis without harm.

Remark 5.2.5.

1. Let us present a way to interpret our optimal signal process ˆ

ξdefined in

Definition 5.2.4 which can be seen as a generalization to the description of

the optimal signal process in Chapter 4, Section 4.2, and its representation

in Theorem 4.2.2, equation (4.10). For ease of presentation and to avoid

unnecessary technicalities, let us assume here that the convergence in (5.15)

also holds in L1(P)and that ν∈L1(P⊗dt)(these assumptions merely simplify

the justification of the representation in (5.24) below; cf. Lemma 5.5.2 in

Section 5.5). Further, since LT>0on the set {0< η ≤+∞} due to Lemma

5.2.3, note that E[LT]= 0 (unless we are in the special case where η= 0

P-a.s.). Then, by defining the weight process (wt)0≤t<T via

wt≜E[LT|Ft]

(0 ≤t < T)(5.22)

as well as the measure Q≪Pon (Ω,FT)via

≜LT

E[LT],

we may write

ξt=1

E[ΞTLT+∫T

ξre−∫r

κuduνrdr ⏐⏐⏐⏐

Ft]

=wtEQ[ΞT|Ft] + (1 −wt)E⎡

⎣∫T

ξr

e−∫r

κudu

(1 −wt)ct

νrdr⏐⏐⏐⏐

Ft⎤

⎦(5.23)

122

5.3 Stochastic LQ problem with stochastic terminal state constraint

for all 0≤t < T. Recall that the adjoint process (Lt)0≤t<T is a strictly

positive supermartingale by virtue of Lemma 5.2.3. Consequently, the weight

process satisfies

0≤wt<1P-a.s. for all 0≤t < T

(cf. Lemma 5.5.2 below for the strict right inequality). Moreover, we have

the identity

E⎡

⎣∫T

e−∫r

κudu

(1 −wt)ct

νrdr⏐⏐⏐⏐

Ft⎤

⎦= 1 dP⊗dt-a.e. on Ω×[0, T)(5.24)

(again, due to Lemma 5.5.2 below). That is, loosely speaking, similar to

the representation in (4.10) above, the optimal signal process ˆ

ξin (5.23) can

be considered as a convex combination of a weighted average of expected

future target positions of ξand the expected terminal position ΞT, computed

under the auxiliary measure Q, where the weight shifts gradually towards the

ultimate terminal position ΞTas t↑T, provided that η > 0. Indeed, by the

definition of the weight process in (5.22), martingale convergence theorem

and the convergence of the process Lin Lemma 5.2.3, we have

∃lim

t↑Twt= 1 on the set {0< η ≤+∞}.

Note that in the special case η= 0 P-a.s. we would have LT= 0 P-a.s. due to

Lemma 5.2.3 and hence, similar to the expression in (4.7) in Theorem 4.2.1

above, the simpler representation

ξt=E⎡

⎣∫T

ξr

e−∫r

κudu

νrdr⏐⏐⏐⏐

Ft⎤

⎦(0 ≤t < T)

with the same property as in (5.24) with w≡0(again, see Lemma 5.5.2

below).

2. Loosely speaking, the optimal signal process ˆ

ξcan be considered as the pro-

cess given by the ratio b/c of the solution processes of the BS(R)DEs in

(5.8) and (5.7). It preserves the desired terminal target position ΞTon the

set {η > 0}but does not possess a well defined terminal value on the set

{η= 0}.

5.3 Stochastic LQ problem with stochastic terminal state

constraint

We return to the stochastic LQ problem with singular stochastic terminal state

constraint introduced in Section 5.1. Recall that for given x∈Rwe want to find a

progressively measurable control u∈L1([0, T], ds)P-a.s. with controlled process

t≜x+∫t

usds (0 ≤t≤T)(5.25)

123

5 General Linear Quadratic Control for Hedging with Temporary Impact

which minimizes

E[∫T

(Xu

t−ξt)2νtdt +∫T

κtu2

tdt +η(Xu

T−ΞT)2].(5.26)

The delicate issue here is that we allow the random penalization parameter ηto

take the value +∞with positive but not necessarily full probability in order to

incorporate the stochastic terminal state constraint Xu

T= ΞTon {η= +∞}. As a

result, the cost term E[η(Xu

T−ΞT)2]needs to be defined in a suitable way.

One possible approach in order to tackle the random singularity at terminal

time Tconsists of performing a truncation in space, that is, looking at a family

of unconstrained variants of the problem in (5.26) where the random penalization

parameter ηis replaced by truncated versions η∧nfor some constants n > 0.

After having solved these auxiliary problems, one can try to pass to the limit

n↑ ∞. In the very specific case ξ≡ΞT= 0, this has been done in Kruse and

Popier (2016a). Recall that the stochastic LQ problem in (5.26) with bounded

penalization parameter η∧n(as well as bounded processes κand ν) has been

solved in Kohlmann and Tang (2002) (within a Brownian framework) and is fully

characterized by the two BS(R)DEs in (5.7) and (5.8) with terminal conditions

cT=η∧nand bT= (η∧n)ΞT, respectively. As shown by, e.g., Kruse and Popier

(2016a) and Graewe et al. (2015), one can indeed let n↑ ∞ in the BSRDE for c.

When ΞT= 0, though, this is not possible for the linear BSDE in (5.8) in general.

Our main idea to tackle and resolve the delicate stochastic terminal state con-

straint consists of performing a truncation in time instead of space. Specifically, we

propose to define the problem in (5.26) as a properly chosen limit of stochastic LQ

problems with terminal times τ < T. The delicate final state penalty η(Xu

T−ΞT)2

is replaced by an appropriate penalization term at each time τ. Of course, it is

natural to replace ηwith cτ, but not clear at all what should replace ΞTin order to

get time consistent penalization terms. As it turns out, the optimal signal process ˆ

at time τgives such a canonical replacement. That is, in light of limt↑Tˆ

ξt= ΞTon

{0< η ≤+∞} in (5.21) as well as limt↑Tct=ηin (5.10), we propose to define the

performance functional as follows:

J(u)≜lim sup

τ↑T

E[∫τ

(Xu

t−ξt)2νtdt +∫τ

κtu2

tdt +cτ(Xu

τ−ˆ

ξτ)2].(5.27)

The limes superior is taken over all sequences of stopping times (τn)n=1,2,... which

P-a.s. converge strictly from below to the terminal time T. The set of admissible

controls is defined to be the domain of J:

U≜{u∈L1(dt)P-a.s. progressively measurable with J(u)<+∞}.(5.28)

Note that, appealing to Fatou’s Lemma as well as (5.10) and (5.21), all controls

u∈Unecessarily satisfy the random terminal state constraint

T= ΞTon the set {η= +∞}.

124

5.4 Main result

The optimization problem we want to solve can now be formulated as

J(u)→min

u∈U.(5.29)

5.4 Main result

We are now ready to state our main theorem. As it turns out, the optimal control

to our stochastic LQ problem in (5.29) with singular stochastic terminal state con-

straint and its corresponding optimal value are fully characterized by the processes

cand ˆ

ξ. First of all, we have to ensure that our set of admissible controls defined

in (5.28) is not empty. In fact, it follows from our analysis below that U=∅if

and only if

E[∫T

(ξt−ˆ

ξt)2νtdt]<+∞and E[∫[0,T )

ctd[ˆ

ξ]t]<+∞,(5.30)

where [ˆ

ξ]denotes the quadratic variation process of the semimartingale ˆ

ξ. In par-

ticular, (5.30) are necessary and sufficient for well posedness of (5.29).

Theorem 5.4.1. Let Assumption 5.2.1 as well as conditions (5.1),(5.2),(5.5)and

(5.19)hold true. Then, we have U=∅if and only if (5.30)is satisfied. In this

case, the optimal control ˆu∈Ufor problem (5.29)with controlled process ˆ

X·≜Xˆu

is given by the feedback law

ˆut=ct

κt(ˆ

ξt−ˆ

Xt)(0 ≤t < T),(5.31)

and the minimal costs are

J(ˆu) = c0(x−ˆ

ξ0)2+E[∫T

(ξt−ˆ

ξt)2νtdt]+E[∫[0,T )

ctd[ˆ

ξ]t].(5.32)

The proof of Theorem 5.4.1 is deferred to Section 5.5 below. Observe that the

feedback law of the optimal control in (5.31) prescribes a reversion towards the

optimal signal process ˆ

ξtrather than towards the current target position ξt. The

reversion speed is controlled by the ratio c/κ. In particular, on the “blow-up”

set {η= +∞} the optimizer reverts with increased urgency towards the optimal

signal ˆ

ξand hence to the ultimate target position ΞTdue to (5.21). This result

generalizes the insights from the constant coefficient case with almost sure terminal

state constraint presented in Chapter 4, Section 4.2.

Due to the integrability conditions in (5.30) the optimal costs J(ˆu)in (5.32) of the

optimizer ˆuin (5.31) are obviously finite. Actually, they represent the generalized

version of the optimal costs from Theorem 4.2.2, expression (4.11), in Section 4.2

above and nicely separate likewise into three intuitively appealing terms making

transparent how the regularity and predictability of the targets ξand ΞTdetermine

the problem’s optimal value. The first term represents the costs due to a possibly

125

5 General Linear Quadratic Control for Hedging with Temporary Impact

suboptimal initial position x. The second term shows how the regularity of the tar-

get process ξfeeds into the overall costs: Targets which are poorly approximated by

the optimal signal process ˆ

ξin the L2(P⊗νtdt)-sense produce higher costs. Finally,

the third term reveals the importance of the optimal signal’s quadratic variation

process [ˆ

ξ]. Referring to the definition of ˆ

ξin (5.20) (cf. also the representation in

(5.23)), the quadratic variation [ˆ

ξ]can be viewed as a measure for the strength of

the fluctuations in the assessment of the average future target positions of ξ, the

terminal position ΞTand the random variable LTwhich involves the outcome of

the penalization parameter ηat time T. With this respect, the second integrability

condition in (5.30) can be interpreted as encoding a condition on the predictability

of the final stochastic target position ΞTas well as the random penalization param-

eter η. Loosely speaking, it ensures that the outcome of the final position ΞTas

well as the “blow-up” event {η= +∞} on which ΞThas to be matched by controls

in Uare not allowed to come as “too big a surprise” at final time T. Note that a

similar condition is also formulated in Chapter 4, Lemma 4.4.4, above. Ankirch-

ner and Kruse (2015) confine themselves to stochastic terminal state constraints of

the form ΞT=∫T

0λtdt for some progressively measurable and suitably integrable

process (λt)0≤t≤Twhich are gradually revealed as t↑T.

Remark 5.4.2.

1. In the case of constant coefficients νt≡ν∈R+,κt≡κ∈R+and η∈[0,+∞]

the BSRDE in (5.9) boils down to a deterministic ordinary Riccati differential

equation on [0, T ]of the form

c′

t=c2

κ−νsubject to cT=η

with explicitly available solutions

ct=⎧

⎪

⎨

⎪

⎩

√νκ √νκ sinh(√ν/κ (T−t))+ηcosh(√ν/κ (T−t))

ηsinh(√ν/κ (T−t))+√νκ cosh(√ν/κ (T−t))0≤η < +∞

√νκ coth(√ν(T−t)/√κ), η = +∞

for all 0≤t≤T. Consequently, the process Lgiven in (5.14) is also just

deterministic and the optimal signal process ˆ

ξin (5.20) can be computed

explicitly (up to the conditional expectation). In particular, we recover the

explicit results from Theorems 4.2.1 and 4.2.2 above. Therein, the first in-

tegrability condition in (5.30) holds true as soon as ξ∈L2(P⊗dt)and the

second condition is replaced by a condition on the terminal position ΞTfor-

mulated in Lemma 4.4.4.

2. In the special case ξ≡0and ΞT= 0 P-a.s., we recover under the specific

dynamics of the controlled state process Xuin (5.25) the result obtained in

Kruse and Popier (2016a), Theorem 3. In this setup, the optimal control u0

126

5.5 Proofs

with controlled process X0is given by

t=−ct

κt

t=−ct

κt

xe−∫t

κudu =−xLt

κt

for all 0≤t≤T.

Observe that the adjoint process (Lt)0≤t≤Tdefined in (5.14) is linked to the

optimal control u0via the relation L=−κu0(if we set x= 1). The corre-

sponding optimal costs are given by

J(u0) = c0x2.

In fact, Kruse and Popier (2016a) show that the process (x2ct)0≤t<T with

(ct)0≤t<T satisfying the BSRDE in (5.9) (with a slightly weaker singular ter-

minal condition, recall Remark 5.2.2 3.)) is the value process to this partic-

ular optimization problem (cf. also the Feynman-Kac representation result

in Kohlmann and Tang (2002), Section 3.5, of the solution process to the

BSRDE in (5.9) with bounded η). From a Mathematical Finance point of

view, the process (ct)0≤t<T can therefore be regarded as the optimal liqui-

dation cost process in the sense that ctprovides the minimal costs at time t

with respect to the remaining time to maturity T−tof unwinding one unit

of current asset holdings (x= 1) until Tif the event {η= +∞} occurs. The

process Lis characterized by the corresponding optimal liquidation rate u0

and the price impact process κ. In particular, the terminal value LTrepre-

sents, with reversed sign, the ultimate optimal liquidation rate u0

Tweighted

with the ultimate instantaneous price impact κT.

3. In the case where ηis bounded P-a.s. as well as the processes (νt)0≤t≤Tand

(κt)0≤t≤T, we recover, under the specific dynamics of the controlled state

process Xuin (5.25), the results obtained in Kohlmann and Tang (2002),

Theorem 5.2, established within a Brownian framework. Note, however, that

they characterize the optimal control ˆuin terms of the process cand the

solution process to the linear BSDE in (5.8) wich does not correspond to ˆ

ξ. In

particular, the key role played by the process ˆ

ξwas not observed in Kohlmann

and Tang (2002).

5.5 Proofs

Throughout this section we work under the assumptions of our main result, The-

orem 5.4.1. Its verification relies on a completion of squares argument similar to

Kohlmann and Tang (2002) (cf. also Yong and Zhou (1999) for this method in

solving LQ problems). The following lemma summarizes the key identity for our

verification and illustrates again the usefulness of our signal process ˆ

ξ.

127

5 General Linear Quadratic Control for Hedging with Temporary Impact

Lemma 5.5.1. For all progressively measurable, P-a.s. locally L1([0, T), dt)-integrable

processes u, the cost process

Ct(u)≜∫t

(Xu

s−ξs)2νsds +∫t

κsu2

sds +ct(Xu

t−ˆ

ξt)2(0 ≤t < T)

is a nonnegative, càdlàg local submartingale. It allows for the decomposition

Ct(u) = c0(x−ˆ

ξ0)2+At(u) + Mt(u) (0 ≤t < T)(5.33)

for all 0≤t < T, where

At(u)≜∫t

(ξs−ˆ

ξs)2νsds +∫t

csd[ˆ

ξ]s

+∫t

κs(us−cs

κs(ˆ

ξs−Xu

s))2

ds (5.34)

is a right continuous, nondecreasing, adapted process and

Mt(u)≜∫t

(ˆ

ξ2

s−−(Xu

s−)2)dNs+ 2∫t

cs−

Ls−

(ˆ

ξs−−Xu

s−)d˜

Ms(5.35)

with

Mt≜E[ΞTLT+∫T

ξse−∫s

κuduνsds ⏐⏐⏐⏐

Ft](5.36)

is a local martingale on [0, T).

Proof. Let us expand

ct(Xu

t−ˆ

ξt)2=ct(Xu

t)2−2Xu

tctˆ

ξt+ctˆ

ξ2

t(0 ≤t < T)

and then apply Itô’s formula to each of the resulting three terms. This will be

prepared by computing the dynamics of the processes ˆ

ξ,cˆ

ξand cˆ

ξ2, respectively,

in the following steps 1, 2 and 3. In step 4 we put everything together and derive

our main identity (5.33).

Step 1: We start with computing the dynamics of our optimal signal process

(ˆ

ξt)0≤t<T defined in (5.20). For ease of notation, let us define the process

Yt≜∫t

ξre−∫r

κuduνrdr (0 ≤t≤T).

Observe that YT∈L1(P)due to (5.2). Moreover, recall that ΞTLT∈L1(FT−,P)

by (5.19) so that (5.36) defines a càdlàg martingale on [0, T]. By the definition of

ξin (5.20), we can now express ˆ

ξin terms of Yand ˜

Mvia

ξt=1

E[ΞTLT+∫T

ξre−∫r

κuduνrdr ⏐⏐⏐⏐

Ft]=1

Lt(˜

Mt−Yt)(5.37)

128

5.5 Proofs

for all 0≤t < T. Next, recall the dynamics of Lon [0, T)in (5.16) and note that

∆Lt=−Lt−

ct−

∆Ntand [L]c

t=∫t

s−

d[N]c

s,(5.38)

where [L]cand [N]cdenote the path-by-path continuous parts of the quadratic

variations of [L]and [N], respectively (cf., e.g., Protter (2004), Chapter II.6, for

more details). Hence, applying Itô’s formula as in, e.g., Protter (2004), Theorem

II.32, we obtain

Lt=1

L0−∫t

s−

dLs+∫t

s−

d[L]c

+∑

s≤t(1

Ls−1

Ls−

s−

∆Ls).(5.39)

Using (5.38), the summands in the sum in (5.39) above can be written as

Ls−−Ls

LsLs−−∆Ns

Ls−cs−

=∆Ns

cs−

Ls−−Ls

LsLs−

=(∆Ns)2

Lsc2

s−

=(∆Ns)2

Ls−cs−cs

where we also used ∆cs=−∆Nsand thus the identity 1/Ls=cs−/(Ls−cs)in the

last equality. Hence, together with the dynamics of Lin (5.16) and [L]cin (5.38)

we can rewrite (5.39) as

Lt=1

+∫t

νs

Ls−cs−

ds +∫t

Ls−cs−

dNs

+∫t

Ls−c2

s−

d[N]c

s+∑

s≤t

(∆Ns)2

Ls−cs−cs

.(5.40)

Now, integrating by parts in (5.37) and then using the dynamics of 1/L in (5.40)

gives us

ξt=ˆ

ξ0+∫t

Ls−

(d˜

Ms−dYs) + ∫t

ξs−Ls−d(1

Ls)+[1

L,˜

M]t

=ˆ

ξ0−∫t

(ξs−ˆ

ξs−)νs

cs−

ds +∫t

Ls−

d˜

Ms+∫t

ξs−

cs−

dNs

+∫t

ξs−

s−

d[N]c

s+∑

s≤t

ξs−

cs−cs

(∆Ns)2+[1

L,˜

M]t

,(5.41)

where the quadratic covariation can be computed as

L,˜

M]t

=∫t

Ls−cs−

d[˜

M, N]c

+∑

s≤t(∆˜

Ms∆Ns

Ls−cs−

+(∆Ns)2∆˜

Ls−cs−cs).(5.42)

129

5 General Linear Quadratic Control for Hedging with Temporary Impact

Collecting all the sums in (5.41) together with those in (5.42) yields

∑

s≤t

∆Ns

Ls−cs−cs(cs∆˜

Ms+ ∆Ns∆˜

Ms+ˆ

ξs−Ls−∆Ns)

=∑

s≤t

∆Ns

Ls−cs−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs),(5.43)

where we used the fact that

∆˜

Ms=˜

Ms−˜

Ms−=ˆ

ξsLs−ˆ

ξs−Ls−(5.44)

due to the representation of ˆ

ξin (5.37) and the continuity of Y. Plugging back

(5.43) into (5.41) finally gives us

ξt=ˆ

ξ0−∫t

(ξs−ˆ

ξs−)νs

cs−

ds +∫t

Ls−

d˜

Ms+∫t

ξs−

cs−

dNs

+∫t

ξs−

s−

d[N]c

s+∫t

Ls−cs−

d[˜

M, N]c

+∑

s≤t

∆Ns

Ls−cs−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs).(5.45)

Step 2: Let us now compute the dynamics of cˆ

ξ. Again, integration by parts,

together with the dynamics of ˆ

ξin (5.45), yields

ctˆ

ξt=c0ˆ

ξ0+∫t

cs−dˆ

ξs+∫t

ξs−dcs+[c, ˆ

ξ]t

=c0ˆ

ξ0−∫t

ξsνsds +∫t

ξs−

κs

ds +∫t

cs−

Ls−

d˜

+∫t

ξs−

cs−

d[N]c

s+∫t

Ls−

d[˜

M, N]c

+∑

s≤t

∆Ns

Ls−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs)+[c, ˆ

ξ]t.(5.46)

The quadratic covariation in (5.46) can be computed as

[c, ˆ

ξ]t=−∫t

Ls−

d[˜

M, N]c

s−∫t

ξs−

cs−

d[N]c

−∑

s≤t

∆Ns∆˜

Ls−−∑

s≤t

ξs−(∆Ns)2

cs−

−∑

s≤t

(∆Ns)2

Ls−cs−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs).(5.47)

130

5.5 Proofs

The sums of the jumps in the quadratic covariation in (5.47) can be rewritten (using

again the identity in (5.44) as well as the fact that ∆cs=−∆Ns) as

−∑

s≤t

∆Ns

Ls−cs−cs(∆˜

Mscscs−+ˆ

ξs−∆NsLs−cs+ ∆Nsˆ

ξsLscs−−∆Nsˆ

ξs−Ls−cs)

=−∑

s≤t

∆Ns

Ls−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs).

With this observation, plugging back the quadratic covariation in (5.47) into (5.46),

we simply get

ctˆ

ξt=c0ˆ

ξ0−∫t

ξsνsds +∫t

ξs−

κs

ds +∫t

cs−

Ls−

d˜

Ms.(5.48)

Step 3: Next, we compute the dynamics of cˆ

ξ2. Application of integration by

parts together with the dynamics of ˆ

ξin (5.45) yields

ξ2

t=ˆ

ξ2

0+ 2∫t

ξs−dˆ

ξs+ [ˆ

ξ]t

=ˆ

ξ2

0−2∫t

ξs−(ξs−ˆ

ξs−)νs

cs−

ds + 2∫t

ξs−

Ls−

d˜

Ms+ 2∫t

ξ2

s−

cs−

dNs

+ 2∫t

ξ2

s−

d[N]c

s+ 2∫t

ξs−

Ls−cs−

d[˜

M, N]c

+ 2∑

s≤t

ξs−∆Ns

Ls−cs−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs)+ [ˆ

ξ]t.

Consequently, using once more integration by parts, we obtain

ctˆ

ξ2

t=c0ˆ

ξ2

0+∫t

cs−dˆ

ξ2

s+∫t

ξ2

s−dcs+ [c, ˆ

ξ2]t

=c0ˆ

ξ2

0−2∫t

ξs−(ξs−ˆ

ξs−)νsds + 2∫t

cs−ˆ

ξs−

Ls−

d˜

Ms+ 2∫t

ξ2

s−dNs

+ 2∫t

ξ2

s−

cs−

d[N]c

s+ 2∫t

ξs−

Ls−

d[˜

M, N]c

+ 2∑

s≤t

ξs−∆Ns

Ls−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs)+∫t

cs−d[ˆ

ξ]s

+∫t

ξ2

s−

κs

ds −∫t

ξ2

s−νsds −∫t

ξ2

s−dNs+ [c, ˆ

ξ2]t.(5.49)

131

5 General Linear Quadratic Control for Hedging with Temporary Impact

The final quadratic covariation in (5.49) can be computed as

[c, ˆ

ξ2]t=−2∫t

ξs−

Ls−

d[˜

M, N]c

s−2∑

s≤t

ξs−

Ls−

∆˜

Ms∆Ns

−2∫t

ξ2

s−

cs−

d[N]c

s−2∑

s≤t

ξ2

s−

cs−

(∆Ns)2

−2∑

s≤t

ξs−(∆Ns)2

Ls−cs−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs)+∫t

∆csd[ˆ

ξ]s.(5.50)

Observe that the sum of jumps in (5.50) can be rewritten as

−2∑

s≤t

ξs−∆Ns

Ls−cs−cs(∆˜

Mscscs−+ˆ

ξs−∆NscsLs−

+∆Nsˆ

ξsLscs−−ˆ

ξs−Ls−cs−∆Ns)

=−2∑

s≤t

ξs−∆Ns

Ls−cs(ˆ

ξsLscs−−ˆ

ξs−Ls−cs∆Ns),

where we used once more the identity in (5.44) and ∆cs=−∆Ns. With this

observation, plugging back (5.50) into (5.49), we finally obtain

ctˆ

ξ2

t=c0ˆ

ξ2

0−2∫t

ξs−ξsνsds +∫t

ξ2

s−νsds + 2∫t

cs−ˆ

ξs−

Ls−

d˜

+∫t

ξ2

s−dNs+∫t

csd[ˆ

ξ]s+∫t

ξ2

s−

κs

ds. (5.51)

Step 4: Let us now put together all the computations from the preceding steps.

Specifically, let ube a progressively measurable, P-a.s. locally L1([0, T], dt)-integrable

process with corresponding controlled process Xu. Due to our computations in

(5.48) and (5.51) as well as the fact that Xuis continuous and of finite variation,

we get for all 0≤t < T that

ct(Xu

t−ˆ

ξt)2=ct(Xu

t)2−2Xu

tctˆ

ξt+ctˆ

ξ2

=c0(x−ˆ

ξ0)2+∫t

csd[ˆ

ξ]s−∫t

(Xu

s)2νsds + 2∫t

sνsξsds

−2∫t

csus(ˆ

ξs−Xu

s)ds +∫t

κs

(Xu

s−ˆ

ξs)2ds −2∫t

ξsξsνsds +∫t

ξ2

sνsds

+∫t

(ˆ

ξ2

s−−(Xu

s−)2)dNs+ 2∫t

cs−

Ls−(ˆ

ξs−−Xu

s−)d˜

Ms.(5.52)

Observe that the last two stochastic integrands sum up to Mt(u)defined in (5.35).

132

5.5 Proofs

Furthermore, two completions of squares in the third line of (5.52) yield

ct(Xu

t−ˆ

ξt)2

=c0(x−ˆ

ξ0)2+∫t

csd[ˆ

ξ]s−∫t

(Xu

s)2νsds + 2∫t

sνsξsds

+∫t

κs(us−cs

κs(ˆ

ξs−Xu

s))2

ds +∫t

(ξs−ˆ

ξs)2νsds

−∫t

κsu2

sds −∫t

ξ2

sνsds +Mt(u)

=c0(x−ˆ

ξ0)2+∫t

csd[ˆ

ξ]s−∫t

(Xu

s−ξs)2νsds

+∫t

κs(us−cs

κs(ˆ

ξs−Xu

s))2

ds +∫t

(ξs−ˆ

ξs)2νsds

−∫t

κsu2

sds +Mt(u)

Consequently, we can write

0≤Ct(u) = ∫t

(Xu

s−ξs)2νsds +∫t

κsu2

sds +ct(Xu

t−ˆ

ξt)2

=c0(x−ˆ

ξ0)2+∫t

csd[ˆ

ξ]s+∫t

(ξs−ˆ

ξs)2νsds

+∫t

κs(us−cs

κs(ˆ

ξs−Xu

s))2

ds +Mt(u)

=c0(x−ˆ

ξ0)2+At(u) + Mt(u) (0 ≤t < T)(5.53)

with (At(u))0≤t<T as defined in (5.34). Finally, observe that the process (At(u))0≤t<T

is a right continuous, nondecreasing, adapted process and that (Mt(u))0≤t<T is a

càdlàg local martingale because ˜

Mand Nare local martingales on [0, T)and all

integrands in (5.35) are left continuous (cf., e.g., Protter (2004), Theorem III.33).

Consequently, we have that (Ct(u))0≤t<T is a nonnegative, càdlàg local submartin-

gale.

We are now ready to give the proof of our main Theorem 5.4.1:

Proof of Theorem 5.4.1:First, let us assume that U=∅. For any u∈Uwe

can consider the corresponding cost process Ct(u) = c0(x−ˆ

ξ0)2+At(u) + Mt(u),

0≤t<T, as in (5.33) in Lemma 5.5.1 above. Let (τn)n=1,2,... be a localizing

sequence of stopping times of the local martingale (Mt(u))0≤t<T such that τn↑T

P-a.s. strictly from below as n→ ∞ and (Mt∧τn(u))0≤t<T is a uniformly integrable

martingale for each n(cf., e.g., Protter (2004), Chapter I.6, for more details). Then

133

5 General Linear Quadratic Control for Hedging with Temporary Impact

it holds by definition of our performance functional Jin (5.27) that

∞> J(u)≜lim sup

τ↑T

E[Cτ(u)]

≥c0(x−ˆ

ξ0)2+ lim sup

n→∞ {E[Aτn(u)] + E[Mτn(u)]}

=c0(x−ˆ

ξ0)2

+ lim sup

n→∞ {E[∫τn

(ξs−ˆ

ξs)2νsds +∫τn

csd[ˆ

ξ]s

+∫τn

κs(us−cs

κs(ˆ

ξs−Xu

s))2

ds]}

≥c0(x−ˆ

ξ0)2

+ lim sup

n→∞ {E[∫τn

(ξs−ˆ

ξs)2νsds +∫τn

csd[ˆ

ξ]s]}

=c0(x−ˆ

ξ0)2+E[∫T

(ξs−ˆ

ξs)2νsds]+E[∫[0,T )

csd[ˆ

ξ]s],(5.54)

where we used Doob’s Optional Sampling Theorem as, e.g., in Protter (2004),

Theorem I.16, in order to get E[Mτn(u)] = 0 for all n≥1, and the last equality is

due to monotone convergence. In particular, the computations in (5.54) show that

(5.30) necessarily holds true. In other words, setting

v≜c0(x−ˆ

ξ0)2+E[∫T

(ξs−ˆ

ξs)2νsds]+E[∫[0,T )

csd[ˆ

ξ]s]<∞,(5.55)

we have for all u∈Uthe lower bound

J(u)≥v. (5.56)

Now, let us define the control ˆuwith corresponding controlled process ˆ

X≜Xˆuvia

the feedback law

ˆut=ct

κt

(ˆ

ξt−ˆ

Xt) (0 ≤t < T).

Observe that ˆuis a progressively measurable process and P-a.s. locally L1([0, T), dt)-

integrable. We denote by Ct(ˆu) = c0(x−ˆ

ξ0)2+Mt(ˆu) + At(ˆu),0≤t < T, the

corresponding cost process from Lemma 5.5.1. We will now show that ˆu∈Uand

that ˆuattains the lower bound in (5.56), i.e.,

J(ˆu) = v

finishing our verification argument. Indeed, first note that, by choice of ˆu, we have

At(ˆu) = ∫t

(ξs−ˆ

ξs)2νsds +∫t

csd[ˆ

ξ]s(0 ≤t < T)

134

5.5 Proofs

and, in particular, the relation

v=c0(x−ˆ

ξ0)2+E[AT−(ˆu)] <∞.

Next, since C(ˆu)is a non-negative local submartingale on [0, T)by virtue of

Lemma 5.5.1 above, let us again consider a localizing sequence of stopping times

(ˆτn)n=1,2,... such that ˆτn↑TP-a.s. strictly from below for n→ ∞ and such that

(Mt∧ˆτn(ˆu))0≤t<T is a uniformly integrable martingale for each n. Then, for any

stopping time τ < T P-a.s., application of Fatou’s Lemma and once more Doob’s

Optional Sampling Theorem yields

E[Cτ(ˆu)] = E[lim inf

n→∞ Cτ∧ˆτn(ˆu)] ≤lim inf

n→∞ E[Cτ∧ˆτn(ˆu)]

=c0(x−ˆ

ξ0)2+ lim inf

n→∞ {E[Aτ∧ˆτn(ˆu)] + E[Mτ∧ˆτn(ˆu)]}

=c0(x−ˆ

ξ0)2+ lim inf

n→∞ E[Aτ∧ˆτn(ˆu)]

=c0(x−ˆ

ξ0)2+E[Aτ(ˆu)] ≤c0(x−ˆ

ξ0)2+E[AT−(ˆu)] = v,

where we also used monotone convergence as well as the fact that (A(ˆu)t)0≤t<T is

an increasing process P-a.s. Hence, it holds that

J(ˆu) = lim sup

τ↑T

E[Cτ(ˆu)] ≤v < ∞(5.57)

and thus ˆu∈U. In particular, due to (5.56), we actually have J(ˆu) = vas desired.

It is left to argue that ˆ

XT=x+∫T

0ˆutdt exists P-a.s. Indeed, ˆu∈Uimplies

that E[∫T

0ˆu2

tκtdt]<∞. Application of Cauchy-Schwarz inequality together with

condition (5.1) yields that ˆu∈L1([0, T ], dt)P-a.s.

Finally, let us assume that (5.30) is satisfied. Then, it follows from (5.55) and

(5.57) that ˆu∈U, i.e., U=∅. In other words, condition (5.30) is sufficient.

The final lemma justifies the interpretation in Remark 5.2.5:

Lemma 5.5.2. Let us assume that limt↑TLt=LTin L1(P)and that ν∈L1(P⊗dt).

Then, we have dP⊗dt-a.e. on Ω×[0, T)the representation

ct=E[LTe∫t

κudu +∫T

e−∫r

κuduνrdr ⏐⏐⏐⏐

Ft].(5.58)

Moreover, we have the identity

E⎡

⎣∫T

e−∫r

κudu

(1 −wt)ct

νrdr⏐⏐⏐⏐

Ft⎤

⎦= 1 dP⊗dt-a.e. on Ω×[0, T),(5.59)

where the weight process (wt)0≤t<T defined in (5.22)satisfies 0≤wt<1P-a.s. for

all 0≤t < T .

135

5 General Linear Quadratic Control for Hedging with Temporary Impact

Proof. Recall the dynamics of the process (Lt)0≤t<T in (5.16), i.e.,

Lt=c0−∫t

e−∫r

κuduνrdr −∫t

e−∫r

κududNr(0 ≤t < T).

Hence, for all 0≤t≤s < T we may write

Ls−Lt=−∫s

e−∫r

κuduνrdr −∫s

e−∫r

κududNr.(5.60)

Observe that the stochastic integral in (5.60) is a martingale on [0, T)by Assump-

tion 5.2.1, property (5.11). Thus, applying conditional expectation to the identity

in (5.60) yields dP⊗dt-a.e. on Ω×[0, T)that

E[Ls|Ft]−Lt=−E[∫s

e−∫r

κuduνrdr ⏐⏐⏐⏐

Ft].(5.61)

Passing to the limit s↑Tin (5.61) we obtain, due to monotone convergence and

due to the assumption that Lsconverges in L1(P)to LT, the representation

Lt=E[LT+∫T

e−∫r

κuduνrdr ⏐⏐⏐⏐

Ft]dP⊗dt-a.e. on Ω×[0, T).(5.62)

In other words, using that Lt=cte−∫t

κudu, we can write

ct=E[LTe∫t

κudu +∫T

e−∫r

κuduνrdr ⏐⏐⏐⏐

Ft]dP⊗dt-a.e. on Ω×[0, T)

as desired for (5.58). Finally, by definition of the weight process (wt)0≤t<T in (5.22)

together with the identity in (5.58) we can write

wt=E[LT|Ft]

=e∫t

κudu

E[LT|Ft] = 1

E[e∫t

κuduLT⏐⏐⏐Ft]

ct(ct−E[∫T

e−∫r

κuduνrdr ⏐⏐⏐Ft])

= 1 −1

E[∫T

e−∫r

κuduνrdr ⏐⏐⏐Ft]for all 0≤t < T, (5.63)

which yields our claim (5.59). In particular, representation (5.63) also reveals that

0≤wt<1P-a.s. for all 0≤t<T. Finally, note that the representation in (5.58)

also holds true in the case η= 0 P-a.s. which implies LT= 0 P-a.s. by Lemma

5.2.3.

5.6 Appendix

Let us briefly discuss the integrability condition (5.13) in our standing Assumption

5.2.1. This condition is not regularly discussed in the BSRDE literature and thus

136

5.6 Appendix

calls for a verification in some sufficiently generic setting. So let us place ourselves

in the context of Kruse and Popier (2016a). We assume that the underlying filtra-

tion (Ft)0≤t≤Tis quasi-left continuous and we let (νt)0≤t≤Tbe a nonnegative and

(κt)0≤t≤Ta strictly positive progressively measurable process which satisfy

E[∫T

(κ2

t+ν2

t)dt]<∞and E[∫T

κt

dt]<∞.(5.64)

Under these conditions, it follows from Corollary 1 in Kruse and Popier (2016a)

that there exists a pair (c, N)which satisfies the BSRDE dynamics in (5.9) on

[0, T)with the slightly weaker singular terminal condition

lim inf

t↑Tct≥ηP-a.s. (5.65)

for some η≥0P-a.s. with P[η= +∞]>0instead of (5.10) (but this is not crucial

for our present discussion; for (5.10) to hold true, further assumptions on ηand

(κt)0≤t≤Tare needed; cf. Popier (2016)). As shown in Kruse and Popier (2016a),

the solution pair (c, N)satisfies (5.11), that is,

E[sup

s∈[0,t]|cs|2+ [N]t]<∞for all 0≤t < T (5.66)

and for any t∈[0, T)we have the estimates

0≤ct≤1

(T−t)2E[∫T

(κs+ (T−s)2νs)ds ⏐⏐⏐⏐

Ft]P-a.s.;(5.67)

cf. Proposition 3 and Remark 4 as well as Corollary 1 in Kruse and Popier (2016a)

with p= 2. In addition to that, we can derive the following lower bound:

Lemma 5.6.1. For all t∈[0, T)we have

ct≥E⎡

⎣

∫T

κsds +1

η⏐⏐⏐⏐

Ft⎤

⎦P-a.s. (5.68)

Proof. We will adopt the same idea as in the proof of Lemma 11 in Popier (2006)

in the case κ≡1(and ν≡0). For all n≥1we define the processes

Γn

t≜E⎡

⎣

∫T

κsds +1

η∧n⏐⏐⏐⏐

Ft⎤

⎦(0 ≤t≤T).

Note that Γnis well defined because the term in the conditional expectation is

bounded by n. Moreover, we have pathwise the identity

∫T

κsds +1

η∧n

=η∧n−∫T

κs⎛

⎝

∫T

κudu +1

η∧n⎞

⎠

ds (0 ≤t≤T).

137

5 General Linear Quadratic Control for Hedging with Temporary Impact

Thus, the process Γnverifies

Γn

t=E⎡

⎢

⎣η∧n−∫T

κs⎛

⎝

∫T

κudu +1

η∧n⎞

⎠

ds ⏐⏐⏐⏐⏐

Ft⎤

⎥

⎦

=E[η∧n−∫T

κs((Γn

s)2+Un

s)ds ⏐⏐⏐Ft](0 ≤t≤T)

with adapted process Ungiven by

s≜E⎡

⎢

⎣

(∫T

κudu +1

η∧n)2⏐⏐⏐⏐⏐

Fs⎤

⎥

⎦−(Γn

s)2(0 ≤s≤T).

Now, observe that Un

s≥0for all 0≤s≤Tdue to Jensen’s inequality. Thus, the

comparison result in Kruse and Popier (2016b), Proposition 4, together with the

construction of the solution process (ct)0≤t<T via a truncation procedure in Kruse

and Popier (2016a), finally yields that for all 0≤t < T we have

ct≥E⎡

⎣

∫T

κsds +1

η∧n⏐⏐⏐⏐

Ft⎤

⎦P-a.s.

Appealing to the monotone convergence theorem for n→ ∞ yields the desired

result.

From now on, still within the context of Kruse and Popier (2016a), let us for sim-

plicity further confine ourselves to a Brownian framework (that is, the underlying

filtration (Ft)0≤t≤Tis the completed filtration generated by a Brownian motion)

and let us make besides (5.64) the following additional assumptions on (νt)0≤t≤T,

(κt)0≤t≤Tand η: We assume that the process (κt)0≤t≤Tis bounded from below and

above, i.e., it holds that

0< k ≤κt≤K < ∞(0 ≤t≤T)(5.69)

for some constants k, K ∈R. Moreover, we assume that the process

T−tE[∫T

(T−s)2νsds ⏐⏐⏐⏐

Ft]≤C(0 ≤t≤T)(5.70)

for some constant C < ∞. Finally, we assume that there exists a constant ε > 0

such that

P[ε≤η≤+∞] = 1.(5.71)

Observe that condition (5.71) implies in particular that ct>0P-a.s. for all t∈

[0, T)by virtue of Lemma 5.6.1. Then, the following holds true.

138

5.6 Appendix

Lemma 5.6.2. Under the conditions (5.64),(5.69),(5.70)and (5.71)the solution

process (ct)0≤t<T to the BSRDE in (5.9)on [0, T)with singular terminal condition

(5.65)satisfies

∫T

d⟨c⟩t

<∞on the set {η= +∞},

i.e., condition (5.13)holds true.

Proof. We extend the proof of Proposition 10 in Popier (2006) done for the specific

case κ≡1and ν≡0to our more general setting by using the upper and lower

bounds of the process (ct)0≤t<T in (5.67) and (5.68). First, note that assumptions

(5.69) and (5.71) imply for the lower bound in (5.68) that

ct≥kε

(T−t)ε+k(0 ≤t < T).(5.72)

Concerning the upper bound in (5.67), we obtain due to (5.69) and (5.70)

ct≤K+const

T−t(0 ≤t < T).(5.73)

Since the process cis bounded from below on [0, T], we can apply Itô’s formula

on [0, T −δ]for some 0< δ < T to the process √(T−t)ct. Using the BSRDE

dynamics of cin (5.9), we obtain

0≤√(T−t)ct

=√Tc0+∫t

0(√T−s

2√cs(c2

κs−νs)−√cs

2√T−s)ds

−1

8∫t

√T−s

c3/2

d⟨c⟩s−1

2∫t

√T−s

√cs

dNs

=√Tc0+1

2∫t

√T−s√cs

κs(cs−νsκs

cs−κs

T−s)ds

−1

8∫t

√T−s

c3/2

d⟨c⟩s−1

2∫t

√T−s

√cs

dNs(0 ≤t≤T−δ)

and hence

8∫T−δ

√T−s

c3/2

d⟨c⟩s+1

2∫T−δ

√T−s

√cs

dNs

≤√Tc0+1

2∫T−δ

√T−s√cs

κs(cs−νsκs

cs−κs

T−s)ds (5.74)

for all 0< δ < T. Observe that due to the boundedness of cin (5.72) and (5.73)

and κin (5.69) as well as the integrability assumption on νin (5.64) it holds for

139

5 General Linear Quadratic Control for Hedging with Temporary Impact

all 0< δ < T that

E[∫T−δ

√T−s√cs

κs⏐⏐⏐⏐

cs−νsκs

cs−κs

T−s⏐⏐⏐⏐

ds]

≤constE[∫T−δ

0⏐⏐⏐⏐

cs−νsκs

cs−κs

T−s⏐⏐⏐⏐

ds]

≤const(E[∫T−δ

csds]+E[∫T−δ

νsκs

ds]+E[∫T−δ

κs

T−sds])<∞.

Hence, by using the upper bound on cin (5.67) and Fubini’s Theorem, we can

compute

E[∫T−δ

0(cs−νsκs

cs−κs

T−s)ds]≤E[∫T−δ

0(cs−κs

T−s)ds]

≤E[∫T−δ

0(1

(T−s)2E[∫T

(κu+ (T−u)2νu)du ⏐⏐⏐⏐

Fs]−κs

T−s)ds]

≤E[∫T−δ

(T−s)2(∫T

κudu)ds −∫T−δ

κs

T−sds]

+E[∫T−δ

(T−s)2(∫T

(T−u)2νudu)ds].(5.75)

Using once more Fubini’s Theorem and the fact that κt≤Kfor all 0≤t≤T, we

get for the first expectation in (5.75) the estimate

E[∫T−δ

(T−s)2(∫T

κudu)ds −∫T−δ

κs

T−sds]

=E[∫T−δ

κu

T−udu +∫T

T−δ

κu

δdu −1

T∫T

κudu −∫T−δ

κs

T−sds]

≤K. (5.76)

Concerning the second expectation in (5.75), application of Fubini’s Theorem yields

E[∫T−δ

(T−s)2(∫T

(T−u)2νudu)ds]

≤E[∫T−δ

(T−u)νudu +δ∫T

T−δ

νudu].(5.77)

Consequently, taking expectation in (5.74) and using that the stochastic integral

with respect to Nin (5.74) is a true martingale on [0, T −δ]due to (5.72) and

(5.66), we obtain together with the estimates in (5.76) and (5.77) the upper bound

8E[∫T−δ

√T−s

c3/2

d⟨c⟩s]

≤√Tc0+const (K+E[∫T−δ

(T−u)νudu +δ∫T

T−δ

νudu]).

140

5.6 Appendix

Passing to the limit δ↓0we get with monotone convergence

E[∫T

√T−s

c3/2

d⟨c⟩s]

≤8(√Tc0+const (K+E[∫T

(T−u)νudu]))<∞,(5.78)

due to (5.64). Now, using (5.68), observe that we can further estimate the process

(ct)0≤t<T from below by

cs≥E⎡

⎣

∫T

κudu +1

η⏐⏐⏐⏐

Fs⎤

⎦≥E⎡

⎣

∫T

κudu +1

1{η= +∞} ⏐⏐⏐⏐

Fs⎤

⎦

=E⎡

⎣

∫T

κudu1{η= +∞} ⏐⏐⏐⏐

Fs⎤

⎦≥k

T−sE[1{η= +∞} |Fs].

Plugging back this lower bound into the left hand side of (5.78) and using optional

projection, we get

∞>E[∫T

√T−s

c3/2

d⟨c⟩s]=E[∫T

√T−s

√csd⟨c⟩s]

≥√kE[∫T

E[1{η= +∞} |Fs]d⟨c⟩s]=√kE[∫T

1{η= +∞} d⟨c⟩s]

=√kE[1{η= +∞} (∫T

d⟨c⟩s)],

which yields the desired result.

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