Optimal execution with multiplicative price impact and incomplete information on the return
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Dammann, Felix; Ferrari, Giorgio Working Paper Optimal execution with multiplicative price impact and incomplete information on the return Center for Mathematical Economics Working Papers, No. 663 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Dammann, Felix; Ferrari, Giorgio (2022) : Optimal execution with multiplicative price impact and incomplete information on the return, Center for Mathematical Economics Working Papers, No. 663, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-29614880 This Version is available at: https://hdl.handle.net/10419/273039 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
663 February 2022 Optimal Execution with Multiplicative Price Impact and Incomplete Information on the Return Felix Dammann and Giorgio Ferrari Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en
OPTIMAL EXECUTION WITH MULTIPLICATIVE PRICE IMPACT AND INCOMPLETE INFORMATION ON THE RETURN FELIX DAMMANN AND GIORGIO FERRARI Abstract. We study an optimal liquidation problem with multiplicative price impact in which the trend of the asset’s price is an unobservable Bernoulli random variable. The investor aims at selling over an infinite time-horizon a fixed amount of assets in order to maximize a net expected profit functional, and lump-sum as well as singularly continuous actions are allowed. Our mathematical modelling leads to a singular stochastic control problem featuring a finite-fuel constraint and partial observation. We provide the complete analysis of an equivalent three-dimensional degenerate problem under full information, whose state process is composed of the asset’s price dynamics, the amount of available assets in the portfolio, and the investor’s belief about the true value of the asset’s trend. The optimal execution rule and the problem’s value function are expressed in terms of the solution to a truly two-dimensional optimal stopping problem, whose associated belief-dependent free boundary btriggers the investor’s optimal selling rule. The curve bis uniquely determined through a nonlinear integral equation, for which we derive a numerical solution allowing to understand the sensitivity of the problem’s solution with respect to the relevant model’s parameters. Keywords: optimal execution problem, multiplicative price impact, singular stochastic control, partial observation, optimal stopping. MSC2020 subject classification: 93E20, 93C41, 49L20, 91G80 JEL classification: G11, C61 1. Introduction In this paper, we consider an investor who possesses a fixed amount of assets and aims at selling them on the market. We assume that the investor faces the issue of causing an adverse price reaction, so that fast selling depresses the stock price, while splitting the order over time may take too long. This problem – also known as the optimal execution problem in algorithmic trading – thus deals with the question of how to trade optimally in order to maximize a given profit, and therefore of how to determine the time as well as the size of the order. Dating back to the early works of Bertsimas and Lo [9], Almgren and Chriss [1] and Almgren [2], the study of optimal execution strategies has received much attention and resulted in a series of important contributions in various settings, which, amongst other modeling features, can be distinguished with respect to the considered type of price impact: Additive or multiplicative. A comprehensive discussion on the latter class of models can be found in Guo and Zervos [41], who also point out that models with multiplicative price impact seem to be more natural since they ensure prices to remain positive. Amongst those works dealing with multiplicative price impact, let us mention Bertsimas et al. [10] for a discrete-time framework, Forsyth et al. [35] for a continuous-time model `a la Black-Scholes, Guo and Zervos [41] and Becherer et al. [5] for settings involving singular stochastic controls. A common feature in the literature is the assumption that the investor has full information on the trend of the asset. This, however, can be a strong requirement. As pointed out by Ekstr¨om and Lu [27], a statistical estimation of the drift is not an efficient procedure, and obtaining a reasonable precision would need data of decades or even centuries under the same market conditions – which is simply not feasible in reality (see also the discussion in Rogers [54], Section 4.2). In some cases, such as initial public offerings, this price history does not even exist. Date: February 24, 2022. 1
2 DAMMANN AND FERRARI To account for this fact, we propose a model of optimal execution with multiplicative price impact in which the drift of the stock price dynamics is a random variable, which is not directly observable by the investor. Through monitoring the evolution of the price on the market, the investor is able to update her belief regarding the drift value. However, such observation is noisy as the investor cannot perfectly distinguish whether price variations are caused by the drift or the stochastic driver of the underlying dynamics. From a mathematical point of view, our model leads to a finite-fuel singular stochastic control problem under partial observation, and we investigate how the presence of incomplete information influences the selling strategy of the investor. In particular, we show that the flow of incoming information – through the observation of the asset’s market price – has a direct effect on the optimal execution rule. Indeed, differently to the case of full information treated in Guo and Zervos [41], the decision to sell is no longer triggered by a constant critical price, but the execution threshold changes dynamically depending on the investor’s current belief on the future trend of the asset. Our results show that the optimal execution strategy is in fact determined by a boundary that is increasing in the belief towards the larger drift value, underlying the intuition that the decision maker chooses to delay selling assets if future prices are expected to increase. In this regard, our work relates to the bunch of economic and financial literature where questions of optimal decision-making under partial observation have been considered; amongst a large number of contributions, we refer to the seminal papers on portfolio selection by Detemple [25] and Gennotte [40]; to Veronesi [59] for an equilibrium model with uncertain dividend drift; to Sass and Haussmann [55] for a terminal-wealth portfolio optimization problem, and to the more recent Colaneri et al. [15] for an optimal liquidation problem with rate strategies and partial observation. Notably, the recent Drissi [26] and Bismuth et al. [11] incorporate Bayesian learning in a model of multi-asset optimal execution, although restricting the agent to absolutely continuous (regular) controls. Furthermore, we contribute to those models dealing with problems of optimal stopping and singular stochastic control. To name just a few recent works, Callegaro et al. [13] for public debt control, De Angelis [21] and D´ecamps and Villeneuve [23] for dividend payments, D´ecamps et al. [22] for investment timing, Ekstr¨om and Lu [27] as well as Ekstr¨om and Vaicenavicius [28] for asset liquidation, Federico et al. [30] for inventory management, Johnson and Peskir [43] for quickest detection, Gapeev [37] for the pricing problem of perpetual commodity equities, and Gapeev and Rodosthenous [38] for a zero-sum optimal stopping games associated with perpetual convertible bonds. Our model, approach and overview of the mathematical analysis. We now discuss the mathematical modeling and analysis. We consider an investor holding a fixed amount yof assets in her portfolio. In absence of the investor’s actions, the stock price evolves according to a geometric Brownian motion dSt=βStdt +σStdWt, where Wis a standard Brownian motion and σ > 0 a constant volatility parameter. Furthermore, the price process exhibits a random future trend β, which is however unknown to the decision maker, and is assumed to be a random variable, independent of the Brownian noise, taking two values β0< β1, for some β0, β1∈Rand β0<0. The decision maker is able to sell the assets on the market over an infinite time horizon, and we denote by ξtthe cumulative amount of assets liquidated up to time t. Consequently, the remaining assets in the portfolio follow the deterministic dynamics Yξ t=y−ξt. Clearly, it has to be ξt≤yat any time t≥0 (finite-fuel constraint), since no more than the initial amount of assets can be sold. As anticipated, we assume that the investor causes an adverse price reaction upon selling, which, following Guo and Zervos [41], we assume to be of multiplicative type. Hence, the controlled asset’s price evolves as dSξ t=βSξ tdt +σSξ tdWt−αSξ t◦dξt, Sξ 0−=s > 0, where α > 0 denotes the parameter of price impact, and the operator ◦is defined as in (2.3) below so to take care of the continuous and jump components of any admissible selling strategy ξ. Notice that the multiplicative price impact structure allows to express the asset’s price process as Sξ= exp(Xξ).
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 3 Here, Xξis then a linearly controlled drifted Brownian motion with volatility σ > 0 and drift value µ=β−1 2σ2. The investor aims at maximizing the total expected discounted reward upon selling, net of transaction costs; that is, sup ξ EhZ∞ 0 e−rteXξ t−κ◦dξti, where the optimization is taken over a suitable admissible class of selling strategies. The latter is a finite-fuel singular stochastic control problem under partial observation. By relying on classical filtering techniques (cf. Shiryaev [57], Section 4.2), we begin by determining an equivalent Markovian problem – the so-called separated problem – under full information (see Fleming and Pardoux [32] as a classical reference on the separated problem). To this end, we introduce the process Π, according to which the investor can update her belief regarding the true value of the drift. This is done by observing the evolution of the process X0(denoting the uncontrolled version of the process Xξ), whose natural filtration FX0 tmodels the overall information available up to time t. More precisely, after forming a prior π:= P(µ=µ1)∈(0,1), the investor dynamically updates her belief upon the arrival of new information through observing the process X0, so that the belief process is given by Πt=P(µ=µ1| FX0 t). Notice that a value of Π close to 1 indicates a strong belief towards the larger value of the drift, while Π close to 0 displays a strong belief in the lower value. Hence, we expect the investor to change the liquidation strategy dynamically and not solely base it on the current price on the market, but also on the present belief at that time. The separated problem turns out to be a three-dimensional degenerate finite-fuel singular stochastic control problem, so that obtaining explicit solutions through a traditional “guess-and-verify approach” is in general not feasible.1 In order to tame the multidimensional nature of the resulting optimal execution problem under full information, we then follow a direct approach which hinges on the study of a suitable optimal stopping problem with value v, that we expect to be associated to the singular stochastic control problem. This method was studied and refined by many authors such as Beneˇs et al. [6], El Karoui and Karatzas [29], and Karatzas and Shreve [45], or De Angelis [21], De Angelis et al. [19] and [20], and Guo and Tomecek [42] for more recent contributions. The optimal stopping problem, which involves the underlying two-dimensional diffusion (X0,Π) taking values in R×(0,1), can be interpreted as an optimal selling problem and exhibits a structure similar to that of the problem treated by D´ecamps et al. [22] (see also Ekstr¨om and Lu [27] for a parabolic version). However, the specific choice of possible drift values assumed in D´ecamps et al. [22] allows for the explicit construction of a solution to the related variational inequality, which is instead not possible in our context. We then solve the optimal stopping problem by relying on techniques from free-boundary theory (as illustrated in the monography by Peskir and Shiryaev [53]) and first show that the optimal stopping rule is characterized through a belief-dependent free boundary a(π) for π∈(0,1). However, the coupled dynamics of the underlying processes X0and Π, as well as the fact that they are driven by the same Brownian motion, makes a further study of the free boundary and the value function vnot feasible. It is for that reason we proceed by deriving two equivalent representations of the optimal stopping problem, which allow for a thorough analysis. First, via a change of measure, the state process (X0,Π) is transformed into (X0,Φ) taking values in R×(0,∞) and with decoupled dynamics. Here, the process Φ is the so-called “likelihood ratio”. Again, we can express the optimal stopping strategy in terms of a free boundary ϕ7→ b(ϕ), which results from a simple transformation of the boundary π7→ a(π). Second, we pass yet to another formulation by deriving the intrinsic parabolic formulation of the stopping problem in coordinates (X0, Z), in which the process Znow follows purely deterministic dynamics and takes values in R. Even though the monotonicity result 1A “guess-and-verify approach” is applicable if we take β0=−β1, which indeed allows for a dimension reduction; see, e.g., D´ecamps and Villeneuve [23]. In this paper, however, we do not consider any relation amongst β0and β1 other than β0< β1.
4 DAMMANN AND FERRARI of the associated free boundary z7→ c(z) is certainly not trivial to derive and calls for a rigorous technical analysis, it is in this formulation that we are able to provide further regularity results of cand of the transformed optimal stopping value function bv. In fact, borrowing arguments from De Angelis [21], suitably adapted to the present setting, we achieve a global regularity of bv, namely bv∈C1(R2). The latter result also allows proving bvxx ∈L∞ loc(R2), and finally obtaining a nonlinear integral equation uniquely solved by the optimal stopping boundary c. It is worth mentioning that such a characterization can be traced back to both optimal stopping boundaries band aand is thus tantamount to a complete specification of the optimal stopping rule in the original (x, π)-coordinates. The thorough analysis developed for the optimal stopping problem is then exploited in order to identify an optimal execution strategy. In fact, the derived regularity results for bvpermits us to prove a verification theorem, that identifies an optimal execution rule and shows that the optimal stopping value function vindeed coincides with a directional derivative of the separated problem’s value function V. Namely, we show that V(x, y, π) := 1 αZx x−αy v(x0, π)dx0,(x, y, π)∈R×(0,∞)×(0,1). Notice, that if α↓0, one finds V(x, y, π) = yv(x, π), which is the value of the problem in which the investor has no market impact. The optimal execution rule can be thought of as a “myopic one”. Indeed, it prescribes to sell assets as if the size of the investor’s portfolio were infinite, and to stop selling once the asset’s inventory is depleted (see also Karatzas [46] and El Karoui and Karatzas [29]). The optimal selling rule involves lump-sum executions (whenever the asset’s price is sufficiently large), that could eventually result into an immediate depletion of the portfolio (if the initial portfolio size is sufficiently small). However, for relatively large portfolios, an initial lump-sum selling is followed by a policy of oblique reflection type. This is triggered by the belief-dependent boundary ϕ7→ b(ϕ) (equivalently, π7→ a(π)). Notably, given that all the transformations developed for the resolution of the optimal stopping problem are one-to-one and onto, the integral equation for the boundary z7→ c(z) yields an integral equation for ϕ7→ b(ϕ), and therefore a complete characterization of the optimal execution rule. In order to provide insights about the sensitivity of the optimal decision mechanism of the investor with respect to the model’s parameters, we develop a recursive numerical scheme, which relies on an application of the Monte-Carlo method. Our contributions. Overall, we believe that the contributions of this paper are the following. Even though the literature on optimal execution problems is extensive (see, to name just a few, Almgren and Chriss [1], Almgren [2], Becherer et al. [5], Bertsimas and Lo [9], Bertsimas et al. [10], Colaneri et al. [15], Gatheral and Schied [39], Guo and Zervos [41], Moreau et al. [50], Schied and Sch¨oneborn [56]), the combination of incomplete information on the future price trend while allowing for lumpsum as well as singularly continuous executions constitutes a novelty. Furthermore, the present study on the optimal execution strategy complements as well as extends the literature on problems with a similar structure under full information. As a matter of fact, the derived optimal execution rule exhibits a broader structure and prescribes to take actions depending on the current belief on the future trend of the asset. From a mathematical point of view, to the best of our knowledge, ours is the first work providing a complete characterization of the value function and of the optimal control rule in a finite-fuel singular stochastic control problem under partial observation (which, in the present setting, is equivalent to a three-dimensional degenerate singular stochastic control problem). Furthermore, we believe that the optimal stopping (selling) problem, studied as a device to characterize the optimal solution of the optimal execution problem, is of interest of its own. By performing a thorough analysis on the regularity of (a transformed version of) its value function and free boundary, we are able to provide a complete characterization of the optimal selling rule through a nonlinear integral equation, thus extending the results of the related model studied by D´ecamps et al. [22]. Notice, that an
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 5 integral equation for the free boundary has been obtained also in Ekstr¨om and Lu [27] and Ekstr¨om and Vaicenavicius [28], though in settings where the parabolic nature of the problem is arising because of an explicit time-dependency. Finally, the probabilistic numerical approach developed for the resolution of the free boundary’s integral equation allows to understand the dependency of the investor’s optimal execution strategy on relevant model’s parameters such as volatility and trend. Organization of the paper. The rest of the paper is organized as follows. In Section 2we present our setting and first preliminary results. In Section 3we investigate the benchmark problem under full information, before we consider a corresponding optimal stopping problem and its optimal boundary in Section 4. In Section 5and 6we derive two equivalent formulations of this problem, which allow for a more thorough study. Eventually, in Section 7, we return to the optimal control problem and characterize the optimal selling rule of the investor. A numerical comparative statics analysis in then carried out in Section 8. 2. Setting and Problem Formulation Let (Ω,F,P) be a complete probability space, rich enough to accommodate a standard onedimensional Brownian motion (Wt)t≥0and an independent random variable βtaking two values β0and β1. We denote by FW:= (FW t)t≥0the filtration generated by (Wt)t≥0augmented by P-null sets of F0. We assume that, in absence of any actions of the investor, the asset’s price on the stock market evolves stochastically according to a geometric Brownian motion dS0 t=βS0 tdt +σS0 tdWt, S0 0=s > 0,(2.1) where σ > 0 is a constant volatility. The investor holds a finite amount y≥0 of assets, which she is able to sell. We identify the cumulative amount of assets sold up to time t≥0, which we denote by ξt, as the investor’s control variable. We denote the natural filtration of any process Zby FZ:= (FZ t)t≥0and hence, the set of admissible execution strategies in this context is given by A(y) := nξ: Ω ×[0,∞) : (ξt)t≥0FS0-adapted, increasing, c`adl`ag, and ξ0−= 0, ξt≤ya.s.o, where the last condition naturally arises from the fact that the investor cannot sell more than the initial amount of assets. Moreover, the remaining assets in the portfolio evolve according to the deterministic dynamics Yξ t=y−ξt, Y ξ 0−=y≥0, where we stress the dependency on the selling strategy ξ. Following Guo and Zervos [41], in our model we assume that the investor’s transactions on the market have a proportional impact on the asset’s price. More precisely, when selling a small amount > 0 of assets at time t, the price exhibits a jump of size ∆St=St−St−=−αSt, for α > 0 denoting the parameter of permanent price impact (see Almgren and Chriss [1], Almgren [2] for early works and Becherer et al. [4], Ferrari and Koch [31], Guo and Zervos [41] for more recent contributions). Hence, a small transaction is such that St= (1 −α)St−≃e−αSt−and, by interpreting a lump-sum sale of ∆ξtshares as a sequence of Nindividual sales of size = ∆ξt/N, we have St=e−αNSt−=e−α∆ξtSt−, for Nlarge enough. It follows that, for any ξ∈ A(y), we can model the controlled asset’s price process by dSξ t=βSξ tdt +σSξ tdWt−αSξ t◦dξt, Sξ 0−=s,(2.2)
6 DAMMANN AND FERRARI where Z· 0 Sξ t◦dξt:= Z· 0 Sξ tdξc t+X t≤· :∆ξt6=0 1 αSξ t−(1 −e−α∆ξt) = Z· 0 Sξ tdξc t+X t≤· :∆ξt6=0 Sξ t−Z∆ξt 0 e−αudu, (2.3) ξcdenotes the continuous part of the process ξ, and ∆ξt:= ξt−ξt−. The solution to (2.2) can be explicitely determined via Itˆo’s formula and it is given by Sξ t=sexp (β−1 2σ2)t+σWt−αξt=S0 texp(−αξt),(2.4) where S0is the solution to (2.1) and we observe that the price impact of selling is additive to the logarithm of the asset’s price. We assume that the investor aims at maximizing the total expected (discounted) profits, net of the total cost of selling, and thus seeks to solve sup ξ∈A(y) EhZ∞ 0 e−rtSξ t−κ◦dξti = sup ξ∈A(y) EhZ∞ 0 e−rtSξ t−κdξc t+X t:∆ξt6=0 e−rt Z∆ξt 0 (Sξ t−e−αu −κ)dui.(2.5) Here, κ > 0 is a proportional transaction cost, which, thinking of Sξ tas the mid-price of the stock at time t, can also be interpreted as a constant bid spread. Notice that the structure of the expected netprofit functional in (2.5) can also be justified through stability results in the Skorokhod M1-topology in probability (see Becherer et al. [5]). Moreover, problem (2.5) has finite value due to ξt≤ya.s. Thanks to (2.4) we have Sξ t= exp(Xξ t), where dXξ t=µdt +σdWt−αdξt, Xξ 0−=x,(2.6) with x:= ln(s) and µ:= β−1 2σ2. In particular, the drift can take two values µi=βi−1 2σ2, i = 0,1. In the following, when needed, we let X0denote the solution to (2.6) with ξ≡0, which is then an arithmetic Brownian motion. Furthermore, we state the following assumption. Assumption 2.1. We have β1> β0and β0<0, which implies µ0<0. The maximization problem (2.5) thus can be rewritten in terms of (2.6) as sup ξ∈A(y) EhZ∞ 0 e−rt eXξ t−κ◦dξti.(2.7) Notice that for a constant non-random drift coefficient, a close variant of this problem was considered and solved by Guo and Zervos [41], who also incorporate the option of buying shares of assets and the constraint that the whole inventory has to be depleted at terminal time. However - due to the presence of incomplete information on the drift of the asset - Problem (2.7) is not of Markovian nature and thus requires a thoroughly different analysis. In order to obtain an equivalent Markovian formulation of (2.7), we rely on classical results from filtering theory, dating back to the contribution of Shiryaev in the context of quickest detection models (see Shiryaev [58] for a survey). To this end, we introduce the belief process Πt:= Pµ=µ1| FX0 t, t ≥0, which reflects the probability at time tthat µ=µ1, conditional on the observations of the price process up to that time (indeed, FX0=FS0). According to this process, the investor is able to update the belief regarding the true value of the drift, based on the arrival of new information by observing the asset’s price evolution on the market. Notice that a large value of Π close to 1 implies a
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 7 strong belief towards the larger drift value µ1, while a low value of Π implies the contrary. It follows (see, e.g., Shiryaev [57], Section 4.2) that the dynamics of Xξ,Π and Yξcan be written as dXξ t= (µ1Πt+µ0(1 −Πt))dt +σdWt−αdξt, Xξ 0−=x∈R, dΠt=γΠt(1 −Πt)dWt,Π0=π∈(0,1), Yξ t=y−ξt, Y ξ 0−=y≥0, (2.8) where γ= (µ1−µ0)/σ is the signal-to-noise ratio and dWt=dX0 t σ−µ0 σ+γΠtdt denotes the innovation process, which is an FX0-Brownian motion on (Ω,F,P). Moreover, π:= P(µ= µ1) reflects the initial subjective belief of the investor regarding the true value of the drift. We do not question the origin of this initial belief, this can either be an instinctive decision or even the result of a constructive approach, for instance by observing the trends of similar assets over the past years. In the new formulation, the process (Xξ, Y ξ,Π) is an FX0-adapted and time-homogeneous Markov process, as it is the unique and strong solution to the system of stochastic differential equations in (2.8). Furthermore, we observe that the drift µis replaced by its conditional estimate and the process Π is a bounded martingale on [0,1] with Π∞∈ {0,1}, as all information will eventually get revealed. Denoting E(x,y,π)[·] = E[·|Xξ 0−=x, Y ξ 0−=y, Π0=π], we can thus reformulate the problem of incomplete information as a so-called separated problem (cf. Bensoussan [8], Chapter 7.1 and Fleming and Pardoux [32]) V(x, y, π) := sup ξ∈A(y) J(x, y, π, ξ),(2.9) with J(x, y, π, ξ) := E(x,y,π)hZ∞ 0 e−rteXξ t−κdξc t+X t:∆ξt6=0 e−rt Z∆ξt 0 (eXξ t−−αu −κ)dui,(2.10) for any (x, y, π)∈R×(0,∞)×(0,1). Notice indeed that Πt∈(0,1) for all t≥0 a.s. if π∈(0,1), while Πt≡π0for all t≥0 a.s. if π0∈ {0,1}. Problem (2.9) is equivalent to (2.5): They share the same value and, because of the uniqueness of the strong solution to (2.8), a control is optimal for (2.5) if and only if it is optimal for (2.9). The Hamilton-Jacobi-Bellman equation. Problem (2.9) takes the form of a three-dimensional singular stochastic control problem with finite-fuel constraint (cf. Baldursson [3], Beneˇs et al. [6], El Karoui and Karatzas [29], Karatzas [44] and Karatzas et al. [48] for early contributions). We start our analysis by providing a heuristic derivation of the dynamic programming equation, that we expect the value function Vto satisfy. To this end, we notice that the investor is faced with two possible actions at initial time. On the one hand, the investor could choose to wait for a short period of time ∆t, not sell any fraction of the assets and then continue with an optimal execution strategy (supposing that one exists). Since this strategy is not necessarily optimal, we obtain V(x, y, π)≥E(x,y,π)e−r∆tV(X∆t, y, Π∆t),(x, y, π)∈R×(0,∞)×(0,1). If we assume that the value function Vhas enough regularity, we can apply Itˆo’s formula, divide by ∆tand invoke the mean value theorem in order to let t→0, so to obtain (LX,Π−r)V≤0. Here, LX,Πdenotes the second-order differential operator, acting on twice-continuously differentiable functions, LX,Π:= 1 2γ2π2(1 −π)2∂ππ +1 2σ2∂xx + (πµ1+ (1 −π)µ0)∂x+σγπ(1 −π)∂xπ.(2.11)
14 DAMMANN AND FERRARI region C2of (5.9) is open. Also, τ∗:= τ∗(x, ϕ) := inf{t≥0 : (Xx t,Φϕ t)∈ S2}is optimal by Peskir and Shiryaev [53], whenever Q-a.s. finite. Furthermore, we define b(ϕ) := inf{x∈R:v(x, ϕ)≤(ex−κ)(1 + ϕ)},(5.11) with inf ∅=∞. In the following lemma, we derive some preliminary properties of the value function (5.8). In light of the relation (5.7) we notice that some of the following results are a direct consequence of Lemma 4.2. Lemma 5.1. The value function vof (5.8) is such that i) 0 ≤v(x, ϕ)≤K1ex(1 + ϕ)for all (x, ϕ)∈R×(0,∞)and some K1>0; ii) x7→ v(x, ϕ)is nondecreasing; iii) ϕ7→ v(x, ϕ)is nondecreasing; iv) (x, ϕ)7→ v(x, ϕ)is locally Lipschitz over R×(0,∞); v) ϕ7→ v(x, ϕ)and x7→ v(x, ϕ)are convex. Proof. Property ii) follows from Lemma 4.2 i), upon using equality (5.7). We prove the remaining claims separately. i) For the lower bound, we notice that {(x, ϕ)∈R×(0,∞) : (ex−κ)<0}⊂C2. Hence, since Φϕ≥0 a.s., we have v(x, ϕ)≥0 for all (x, ϕ)∈R×(0,∞). For the upper bound, we observe that for any stopping time τ EQ (x,ϕ)e−rτ (eXτ−κ)(1 + Φτ)= (1 + ϕ)E(x,π)e−rτ (eXτ−κ) ≤(1 + ϕ)Ee−rτ ex+µ1τ+σWτ≤K1ex(1 + ϕ), for π=ϕ/(1 + ϕ) and the last inequality follows from standard estimates upon using Assumption 4.1. iii) Let ϕ, ϕ0∈(0,∞) with ϕ0> ϕ and notice that Φϕ t=ϕe−1 2γ2t+γBt. For x∈Rand τ∗:= τ∗(x, ϕ) optimal for v(x, ϕ) we have v(x, ϕ0)−v(x, ϕ)≥EQ (x,ϕ0)e−rτ∗eXτ∗−κ1+Φτ∗−EQ (x,ϕ)e−rτ∗eXτ∗−κ1+Φτ∗ =EQe−rτ∗eXx τ∗−κ(ϕ0−ϕ)e−1 2γ2τ∗+γBτ∗≥0, where the last inequality exploits that {(x, ϕ)∈R×(0,∞) : (ex−κ)<0}⊂C2, and the claim follows. iv) Let x, x0∈R,π∈(0,1) and ϕ, ϕ0∈(0,∞). Recall vof (4.3). Again, standard estimates yield |v(x, π)−v(x0, π)| ≤ K1|ex−ex0|,as well as |v(x, ϕ)−v(x, ϕ0)| ≤ K2ex|ϕ−ϕ0|, for some K1, K2>0. Hence, using (5.7), we obtain |v(x, ϕ)−v(x0, ϕ0)|≤|v(x, ϕ)−v(x0, ϕ)|+|v(x0, ϕ)−v(x0, ϕ0)| ≤K1(1 + ϕ)|ex−ex0|+K2ex0|ϕ−ϕ0|,(5.12) and thus the locally-Lipschitz property follows. iv) We first prove convexity regarding ϕ∈(0,∞). For ϕ1, ϕ2∈(0,∞), x∈Rand λ∈(0,1) we set ϕ:= λϕ1+ (1 −λ)ϕ2and obtain v(x, ϕ) = sup τ EQ (x,ϕ)he−rτ (eXτ−κ)(1 + ϕe−1 2γ2τ+γBτ)i ≤sup τ EQhe−rτ (eXx τ−κ)λ(1 + ϕ1e−1 2γ2τ+γBτ)i + sup τ EQhe−rτ (eXx τ−κ)(1 −λ)(1 + ϕ2e−1 2γ2τ+γBτ)i =λv(x, ϕ1) + (1 −λ)v(x, ϕ2),
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 15 and the claim follows. Analogously, upon exploiting the convexity of x7→ ex, one can prove the convexity of x7→ v(x, ϕ). Lemma 5.2. The continuation and stopping region regions as in (5.9)-(5.10)are such that C2={(x, ϕ)∈R×(0,∞) : x<b(ϕ)},S2={(x, ϕ)∈R×(0,∞) : x≥b(ϕ)}. Proof. We proceed similarly to Lemma 4.3. We first notice that the the second-order differential operator associated with the two-dimensional process (X, Φ) is such that LX,Φf=µ0∂xf+1 2σ2∂xxf+1 2γ2ϕ2∂ϕϕf+γϕσ∂xϕf, ∀f∈C2(R×(0,∞)),(5.13) and apply Dynkin’s formula to obtain u(x, ϕ) := v(x, ϕ)−(ex−κ)(1 + ϕ) = sup τ EQ (x,ϕ)hZτ 0 e−rteXt(µ0+1 2σ2−r) + rκ + ΦteXt(µ1+1 2σ2−r) + rκdti.(5.14) For x2> x1and τ∗:= τ∗(x2, ϕ) optimal for v(x2, ϕ) we have u(x1, ϕ)−u(x2, ϕ) ≥EQhZτ∗ 0 e−rteXx2 t−eXx1 t(r−µ0−1 2σ2)+ΦteXx2 t−eXx1 t(r−µ1+1 2σ2)dti ≥0, where the last inequality follows from Xx2≥Xx1Q-a.s. and Assumption 4.1. Hence, for (x1, ϕ)∈ S2 and x2> x1, we obtain 0 ≤u(x2, ϕ)≤u(x1, ϕ) = 0 and the claim follows. It is interesting to notice that there exists a one-to-one correspondence between the continuation regions C1and C2of (4.4) and (5.9) as well as the stopping regions S1and S2of (4.5) and (5.10). Indeed, introducing the diffeomorphism T:= (T1, T2) : R×(0,1) →R×(0,∞),(T1(x, π), T2(x, π)) := x, π 1−π,(5.15) with inverse T−1(x, ϕ) := x, ϕ 1 + ϕ,(x, ϕ)∈R×(0,∞), one has C2=T(C1) as well as S2=T(S1). Furthermore, upon using Lemma 4.3 and Lemma 5.2, we find that b(ϕ) = aϕ 1 + ϕ.(5.16) Due to this explicit relationship between the optimal stopping boundaries, we obtain some first results on bthanks to Lemma 4.4. Lemma 5.3. The boundary b(ϕ)of (5.11) is such that i) ϕ7→ b(ϕ)is nondecreasing on (0,∞); ii) ϕ7→ b(ϕ)is left-continuous; iii) bis bounded by x∗ 0≤b(ϕ)≤x∗ 1for all ϕ∈(0,∞), with x∗ 0and x∗ 1as in Lemma 4.4. The relationship (5.16) and the transformation (5.15) allow us to trace back our results from this section - as well as from the following section - to the initial optimal stopping problem (4.3). Moreover, (5.16) turns out to be valuable in the proof of Lemma 5.3, since proving the monotonicity result i) as well as the boundedness iii) is not straightforward without exploiting the relation between band aand the results of Lemma 4.4.
16 DAMMANN AND FERRARI 6. A Parabolic Formulation Observe that the dynamics of the processes Xand Φ in (5.4) are driven by the same Brownian motion. In order to account for this degeneracy, we pass yet to another formulation of the optimal stopping problem. To this end, we rely on a transformation that reveals the true parabolic nature of the generator LX,Φas in (5.13); i.e. that poses it in its canonical form (cf. Folland [34]). Define T:= (T1, T2) : R×(0,∞)→R2,(T1(x, ϕ), T2(x, ϕ)) := x, σ γln(ϕ)−x,(6.1) for any (x, ϕ)∈R×(0,∞), which is a diffeomorphism with inverse given by T−1(x, z) := x, eγ σ(x+z),(x, z)∈R2.(6.2) With regard to the transformation (6.1) we can introduce the process Zt=σ γln(Φt)−Xt, t ≥0,(6.3) and an application of Itˆo’s formula reveals that its dynamics are given by dZt=−1 2(µ1+µ0)dt, Z0=z:= σ γln(ϕ)−x.(6.4) Furthermore, we can define the transformed version of the value function vof (5.8) via bv(x, z) := vx, eγ σ(x+z)= sup τ EQ (x,z)e−rτ (eXτ−κ)(1 + eγ σ(Xτ+Zτ)),(6.5) for (x, z)∈R2and where now EQ (x,z)[·] = EQ[·|X0=x, Z0=z]. In light of this explicit relationship between the value functions vand bv, we can conclude the following result from Lemma 5.1. Lemma 6.1. The value function bv(x, z)of (6.5)is locally Lipschitz continuous over R2. The associated continuation and stopping region are given by C3:= {(x, z)∈R2:bv(x, z)>(ex−κ)(1 + eγ σ(x+z))},(6.6) S3:= {(x, z)∈R2:bv(x, z)=(ex−κ)(1 + eγ σ(x+z))},(6.7) where C3is open and S3is closed. Furthermore, the global diffeomorphism (6.1) implies that C3= T(C2) as well as S3=T(S2), with C2and S2as in (5.9)-(5.10). Notice that the second-order infinitesimal generator associated to the process (X, Z) is now such that LX,Z f=µ0∂xf+1 2σ2∂xxf−1 2(µ1+µ0)∂zf, ∀f∈C2,1(R2).(6.8) We can rely on standard arguments from classical PDE theory as well as optimal stopping theory (see, e.g., Karatzas and Shreve [47], Section 2.7, Th. 7.7) and obtain the following lemma. Lemma 6.2. The value function bvof (5.7)is the unique classical C2,1-solution to the boundary value problem (LX,Z −r)w= 0 in Rand w|∂R=bv|∂R,(6.9) for LX,Z as in (6.8)and any open set Rsuch that its closure is contained in the continuation region C3of (6.6). In particular, bv∈C2,1(C3). In the following, we aim at investigating the geometry of the state space in the coordinates (X, Z). To this end, we define the generalised inverse of the nondecreasing boundary bby b−1(x) := inf{ϕ∈(0,∞) : b(ϕ)> x},(6.10) such that the continuation region C2of (5.9) rewrites as C2={(x, ϕ)∈R×(0,∞) : b−1(x)< ϕ}.(6.11)
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 17 Since ϕ7→ b(ϕ) is nondecreasing by Lemma 5.3, we observe that (x, z)∈ C3⇐⇒ (x, eγ σ(x+z))∈ C2⇐⇒ eγ σ(x+z)> b−1(x)⇐⇒ z > σ γlog(b−1(x)) −x, and by setting c−1(x) := σ γlog(b−1(x)) −x,(6.12) we can rewrite (6.6) and (6.7) as C3={(x, z)∈R2:z > c−1(x)},S3={(x, z)∈R2:z≤c−1(x)}.(6.13) In contrast to the optimal stopping problems in the formulations (4.3) and (5.8), deriving the monotonicity of the boundary x7→ c−1(x) is not straightforward. Moreover - and differently to related contributions such as Federico et al. [30] - we cannot trace it back to the monotonicity of the boundary bof (5.11), since its generalised inverse b−1is nondecreasing as well, and this does not imply monotonicity of x7→ c−1(x). To this end, we follow and adapt arguments presented in Section 4.4 of De Angelis [21], which studies separately the two cases in which the deterministic process Zas in (6.4) is either increasing (µ0+µ1≥0) or decreasing (µ0+µ1<0). To that end, we state the following assumption, which will be standing from now on. Assumption 6.3. We assume r > 1 2σ2+µ1+(2µ1+σ2)(µ1−µ0) σ2. Remark 6.4. Notice that Assumption 6.3 requires the discount factor rto be larger than the lower bound 1 2σ2+µ1, required in Assumption 4.1 and needed for the well-posedness of the problem. Furthermore, recall that x∗ 0denotes the optimal execution threshold in the case of full information, when the drift is constant and equal to µ0(see Section 3), define ˜x:= log rκ r−1 2σ2−µ1,(6.14) and notice that (LX−r)(ex−κ)≥0, for all x≥˜xand LX=1 2σ2∂xx +µ1∂x. Using the explicit expression of x∗ 0given by (3.6), it can be verified that Assumption 6.3 precisely guarantees x∗ 0>˜x. For the following analysis, it is useful to define bu(x, z) := bv(x, z)−(ex−κ)(1 + eγ σ(x+z)),(6.15) as well as g(x, z) := (LX,Z −r)(ex−κ)(1 + eγ σ(x+z)) =ex1 2σ2+µ0−r+rκ +eγ σ(x+z)ex1 2σ2+µ1−r+rκ,(6.16) and we observe that an application of Dynkin’s formula implies bu(x, z) = sup τ EQ (x,z)hZτ 0 e−rtg(Xt, Zt)dti,(x, z)∈R2.(6.17) Proposition 6.5. Let µ0+µ1≥0. Then there exists a nondecreasing function c:R→Rsuch that the continuation region C3of (6.6)rewrites as C3={(x, z)∈R2:x<c(z)}.(6.18) Proof. Let (x0, z0)∈ S3,x1> x0and notice that (6.13) implies (−∞, z0]×{x0} ∈ S3. Furthermore, Assumption 6.3 guarantees x0> x∗ 0and since the process Zis decreasing, we observe that the process (Xx1, Zz0) crosses the half-line (−∞, z0]×{x0}before reaching the level x∗ 0. Hence, we have Qx1,z0(τ∗< τx∗ 0) = 1, where τx∗ 0:= inf{t≥0 : Xx1 t=x∗ 0}and Qx1,z0(·) = Q(·|X0=x1, Z0=z0), and
18 DAMMANN AND FERRARI Assumption 6.3 implies exp(Xx1 s)(r−1 2σ2−µ1)> rκ for all s∈[0, τ∗). Consequently, (6.16)-(6.17) imply bu(x1, z0)≤0 for all x1> x0, and therefore {z0}×[x0,∞)∈ S3. We can thus define c(z) := inf{x∈R: (x, z)∈ S3}.(6.19) and observe that (6.13) implies that z7→ c(z) is nondecreasing. In order to establish the same result in the case when µ0+µ1<0, we first state the following lemma. Lemma 6.6. We have bvz(x, z) = EQ (x,z)hγ σe−rτ∗(eXτ∗−κ)eγ σ(Xτ∗+Zτ∗) 1 {τ∗<∞}i,(6.20) for all (x, z)∈R2\∂C3and τ∗:= τ∗(x, z). Proof. For (x, z)∈ S3the claim follows immediately, since Q(x,z)(τ∗= 0) = 1. Hence, we let (x, z)∈ C3and for > 0 we obtain bv(x, z +)−bv(x, z)≥EQhe−r(τ∗∧t)bv(Xx τ∗∧t, Zz+ τ∗∧t)−bv(Xx τ∗∧t, Zz τ∗∧t)i ≥EQhe−rτ∗eXx τ∗−κeγ σXx τ∗eγ σZz+ τ∗)−eγ σZz τ∗) 1 {τ∗<t}i +EQhe−rtbv(Xx t, Zz+ t)−bv(Xx t, Zz t) 1 {τ∗>t}i,(6.21) where the first inequality follows from the supermartingale property of e−r(τ∧t)bv(Xx τ∧t, Zz+ τ∧t)tand the martingale property of e−r(τ∗∧t)bv(Xx τ∗∧t, Zz τ∗∧t)tfor τ∗:= τ∗(x, z). Upon employing a change of measure as in Section 5, we find EQ (x,z)e−rt|bv(Xt, Zt)|≤EQ (x,z)e−rt|v(x, eγ σ(Xt+Zt))|≤K1EQ (x,exp( γ σ(x+z))e−rteXt(1 + Φt) =K1(1 + eγ σ(x+z))E(x,π)e−rteXt, where π=eγ σ(x+z)/(1 + eγ σ(x+z)). It is then easy to verify that Assumption 4.1 implies lim t↑∞ EQ (x,z)e−rtbv(Xt, Zt)= 0, and hence, applying dominated convergence in (6.21) as t↑ ∞ yields bv(x, z +)−bv(x, z)≥EQhe−rτ∗eXx τ∗−κeγ σXx τ∗eγ σZz+ τ∗−eγ σZz τ∗ 1 {τ∗<∞}i.(6.22) Similar arguments show bv(x, z)−bv(x, z −)≤EQhe−rτ∗eXx τ∗−κeγ σXx τ∗eγ σZz+ τ∗−eγ σZz τ∗ 1 {τ∗<∞}i,(6.23) and since bv∈C2,1(C3) (cf. Lemma 6.2), dividing (6.22) and (6.23) by and letting ↓0, we obtain the desired result. Proposition 6.7. Let µ0+µ1<0. Under the additional assumption that r > γ 2σ|µ0+µ1|, there exists a nondecreasing function c:R→Rsuch that the continuation region of (6.6)can be written as C3={(x, z)∈R2:x<c(z)}.(6.24) Proof. Let (x, z)∈R2. Notice that x < x∗ 0implies (x0, z)∈ C3for all x0< x and z∈R, because of Lemma 4.4 and since the transformations T1and T1of (5.15) and (6.1), respectively, are the identity; hence, {(x, z) : x < x∗ 0} ⊂ C3. We can thus focus on the case that x≥x∗ 0and distinguish two possibilities: i) bux(x, z)≤0∀x∈(x∗ 0,∞) such that (x, z)∈ C3; ii) ∃x0∈R,x0> x∗ 0such that (x0, z)∈ C3and bux(x0, z)>0.
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 19 In case i), the map x7→ bu(x, z) is decreasing for x∈(x∗ 0,∞) and (x, z)∈ C3. Hence, for any (x, z) in the latter region we obtain (−∞, x]× {z} ∈ C3and the claim follows in the same spirit as in Proposition 6.5. In case ii), we establish a contradiction scheme. As a first step, we show that ii) implies [x0,∞)×{z} ∈ C3, which will then lead to a contradiction. We start by noticing that Lemma 6.2 and (6.17) imply (LX,Z −r)bu(x0, z) = −g(x0, z),(6.25) for (x0, z) as given in ii) above. Since µ0<0 and bux(x0, z)>0 we have µ0bu(x0, z)<0, and thus 1 2σ2buxx(x0, z) = rbu(x0, z)−µ0bux(x0, z) + 1 2(µ0+µ1)buz(x0, z)−g(x0, z)(6.26) > rbu(x0, z) + 1 2(µ0+µ1)buz(x0, z)−g(x0, z). Next, we notice that we can rewrite (6.20) as bvz(x, z) = γ σbv(x, z)−EQ[e−rτ∗(eXx τ∗−κ)],(6.27) and since bvz(x, z) = buz(x, z) + γ σ(ex−κ)eγ σ(x+z)and bv(x, z) = bu(x, z)+(ex−κ)(1 + eγ σ(x+z)), (6.27) gives buz(x, z) + γ σ(ex−κ)eγ σ(x+z)=γ σbu(x, z)+(ex−κ)(1 + eγ σ(x+z))−EQ[e−rτ∗(eXx τ∗−κ)], which is equivalent to buz(x, z) = γ σbu(x, z) + γ σex−κ−EQ[e−rτ∗(eXx τ∗−κ)]. We can thus plug this last equality into (6.26) and obtain 1 2σ2buxx(x0, z) > rbu(x0, z) + 1 2(µ0+µ1)γ σbu(x0, z) + γ σex0−κ−EQ[e−rτ∗(eXx0 τ∗−κ)]−g(x0, z) =r+1 2(µ0+µ1)γ σ(bu(x0, z) + ex0−κ)−1 2(µ0+µ1)γ σEQ[e−rτ∗(eXx0 τ∗−κ)] −g(x0, z) >0, where the last inequality follows from r > γ 2σ|µ0+µ1|and Assumption 6.3, upon noticing that x0> x∗ 0. We deduce that bux(·, z) increases in a right-neighbourhood of x0and repeating arguments for every x > x0yields bux(·, z)>0 on [x0,∞). It follows that bu(·, z) is increasing on [x0,∞) such that [x0,∞)×{z}∈C3and (combining the latter with (6.13)) we have A:= [x0,∞)×[z0,∞)⊂ C3. However, this leads to a contradiction. To see this, let (x, z)∈ A and define τx0:= inf{t > 0 : Xx t≤ x0}. Since t7→ Zz tis increasing, the only possibility for the process (Xx, Zz) to exit Aand thus eventually the continuation region, is by passing through the horizontal line [x0,∞)×{z0}. We thus have τx0≤τ∗Q(x,z)-a.s. and moreover, since µ0<0, the stopping time τx0is finite a.s. Upon using Lemma 5.1 i) and (6.5), it follows that (ex−κ)(1 + eγ σ(x+z))<bv(x, z) = EQ (x,z)he−rτx0bv(Xτx0, Zτx0)i =EQ (x,z)he−rτx0bv(x0, z −1 2(µ0+µ1)τx0)i ≤K1ex0EQ (x,z)he−rτx0i+K1eγ σ(x0+z)ex0EQ (x,z)he−(r−1 2 γ σ|µ0+µ1|)τx0i.
20 DAMMANN AND FERRARI Let now br:= r−γ 2σ|µ0+µ1|>0 and denote φr(resp. φbr) the strictly decreasing solution to 1 2σ2fxx +µ0fx−qf = 0, for q∈ {r, br}. Then, by results on hitting times for one-dimensional diffusions (see, e.g., Borodin and Salminen [12], Ch. II), the above inequality is equivalent to (ex−κ)(1 + eγ σ(x+z))≤K1ex0φr(x) φr(x0)+K1ex0eγ σ(x0+z)φbr(x) φbr(x0),(6.28) which thus holds true for all (x, z)∈ A. Since Ais right-connected, we can let x→ ∞ and notice that (ex−κ)(1 + eγ σ(x+z))→ ∞, while the right hand side of (6.28) decreases to 0 due to the decreasing property of x→φq(x) for qpositive. We thus obtain a contradiction, which concludes our proof. In order to guarantee the existence of the nondecreasing boundary z7→ c(z) in the rest of the paper, we state the following standing assumption. Assumption 6.8. We let Assumption 6.3 hold and that either i) µ0+µ1≥0or ii) µ0+µ1<0and r > γ 2σ|µ0+µ1|. Remark 6.9. Notice that Propositions 6.5 and 6.7 imply that the function x7→ c−1(x)of (6.12)is nondecreasing as well. Moreover, we notice that z > c−1(x)⇐⇒ c(z)> x,(6.29) and hence, the function c−1is the right-continuous inverse of cand thus admits the representation c−1(x) = inf{z∈R:c(z)> x}.(6.30) In light of the connection (6.12)between c−1and b−1(the generalised inverse of the boundary b), equation (6.30)allows us to trace back our results to the formulation of Section 5and then - through the representation (5.16)- to the original setting of Section 4. 6.1. Regularity of the value function and of the optimal stopping boundary. Under the Assumption 6.8 we established the existence of a nondecreasing boundary z7→ c(z), such that R2is split into the continuation region C3of (6.6) and the stopping region S3of (6.7). In the following, we derive some further properties of the optimal stopping boundary and of the value function bvof (6.5). We first state the following result, which will be helpful in the forthcoming analysis. Lemma 6.10. We have buz(x, z)≥0for (x, z)∈ C3. Proof. Because of (6.5) and (6.1), we have that v(x, ϕ) as in (5.8) is such that v(x, ϕ) = bv(x, σ γln(ϕ)− x),(x, ϕ)∈R×(0,∞). Since bvz∈C0(C3) by Lemma 6.2, we then also have vϕ∈C0(C2). Furthermore, ϕ7→ v(x, ϕ) is convex on (0,∞) by Lemma 5.1 iv) and thus also ϕ7→ u(x, ϕ) of (5.14). Then, for (x, ϕ)∈ C2and ϕ0=b−1(x) such that (x, ϕ0)∈∂C2, we obtain (as uϕ∈C0(C2) as well) 0≤u(x, ϕ) = u(x, ϕ)−u(x, b−1(x)) ≤uϕ(x, ϕ)(ϕ−b−1(x)), and ϕ > b−1(x) implies uϕ(x, ϕ)≥0 for (x, ϕ)∈ C2. In light of the relation (6.5) we then obtain buz(x, z)≥0 on C3. Proposition 6.11. The optimal stopping boundary c(z)is such that x∗ 0≤c(z)≤x∗ 1for all z∈R and with x∗ 0and x∗ 1as in Lemma 4.4. Furthermore, we have c∈C(R). Proof. The first part of the claim follows from Lemma 5.3 iii) and by noticing that the transformation T1of (6.1) is the identity. We derive the continuity of z7→ c(z) in two steps. 1) Left-Continuity: Let z0∈Rand zn↑z0as n→ ∞. Since z7→ c(z) is nondecreasing and S3is closed, we obtain limn→∞(c(zn), zn) = (c(z0−), z0)∈ S3, where c(z0−) denotes the left limit of cat z0. The definition of cin (6.19) implies c(z0−)≥c(z0), but since cis nondecreasing, we must have c(z0−) = c(z0) and the claim follows. 2) Right-Continuity: We argue by contradiction and assume there exists z0∈Rs.t. c(z0)< c(z0+).
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 21 Using techniques developed in De Angelis [18], we take c(z0)< x1< x2< c(z0+) and a nonnegative function φ∈C∞ c(x1, x2) such that Rx2 x1φ(x)dx = 1. Recalling (6.25), we have LX,Z bu(x, z)−rbu(x, z) = −g(x, z),(6.31) for (x, z)∈(x1, x2)×(z0,∞). In the following, it is helpful to treat the cases i) µ0+µ1≥0 and ii) µ0+µ1<0 separately. Let us start with i) and recall that buz(x, z)≥0 for xand zas above, due to Lemma 6.10. Integration by parts reveals 0≥ −1 2(µ0+µ1)Zx2 x1buz(x, z)φ(x)dx =Zx2 x1rbu(x, z)−µ0bux(x, z)−1 2σ2buxx(x, z)−g(x, z)φ(x)dx =Zx2 x1rbu(x, z)φ(x) + µ0bu(x, z)φ0(x)−1 2σ2bu(x, z)φ00(x)−g(x, z)φ(x)dx. Hence, employing dominated convergence as z↓z0and using bu(x, z0) = 0, yields 0≥ −Zx2 x1 g(x, z)φ(x)dx > 0,(6.32) where the latter inequality follows from x1, x2≥x∗ 0and Assumption 6.3, which implies x > ˜xfor all x∈[x1, x2] and ˜xas in (6.14). We thus obtain a contradiction and c(z0) = c(z0+). In case ii), we rely on classical results of internal regularity of PDEs (cf. Th. 10 in Chapter 3 of Friedman [36]), which allow to take derivatives in (6.31) with respect to xand have bux∈C2,1(C3) solving (LX,Z −r)bux(x, z) = −gx(x, z),(x, z)∈(x1, x2)×(z0,∞). Then, for z > z0we obtain Zx2 x1(LX,Z −r)bux(x, z) + gx(x, z)φ(x)dx = 0.(6.33) Let Fφ(z) := Rx2 x1buxz(x, z)φ(x)dx. Integration by parts allows to rewrite (6.33) as 1 2|µ0+µ1|Fφ(z) = Zx2 x1rbux(x, z)−1 2σ2buxxx(x, z)−µ0buxx(x, z)−gx(x, z)φ(x)dx =Zx2 x1−rbu(x, z)φ0(x) + 1 2σ2bu(x, z)φ000(x)−µ0bu(x, z)φ00(x)−gx(x, z)φ(x)dx, and using dominated convergence as z↓z0as well as bu(x, z0) = 0 results in Fφ(z0+) = 2 |µ0+µ1|Zx2 x1−gx(x, z0)φ(x)dx ≥p0>0, for some p0, where the second to last inequality again follows from Assumption 6.3. Thus, there exists > 0 such that Fφ(z)≥p0/2 for all z∈(z0, z0+) and we finally obtain 1 2p0≤Zz0+ z0 Fφ(z)dz =Zz0+ z0Zx2 x1buxz(x, z)φ(x)dxdz =−Zx2 x1Zz0+ z0buz(x, z)φ0(x)dzdx =−Zx2 x1 (bu(x, z0+)−bu(x, z0))φ0(x)dx =Zx2 x1bux(x, z0+)φ(x)dx ≤0, where we used bu(x, z0) = 0 as well as bux(x, z)≤0 for x∈[x1, x2] and z > z0(cf. Proposition 6.7). Hence, c(z) = c(z+) for all z∈Rand together with 1) we conclude that z7→ c(z) is continuous. In the next step, we derive the regularity of the value function. Its proof can be found in Appendix A.
22 DAMMANN AND FERRARI Proposition 6.12. The value function bvof (6.5)satisfies bv∈C1(R2)and bvxx ∈L∞ loc(R2). In light of Proposition 6.12, we are able to derive an integral equation for the free boundary c. Let us first recall that by standard arguments, based on the strong Markov property and Proposition 6.12, the value function bvand the free boundary csolve the free-boundary problem (LX,Z −r)bv(x, z)≤0,(x, z)∈R2, (LX,Z −r)bv(x, z) = 0, x < c(z), z ∈R, bv(x, z)≥(ex−κ)(1 + eγ σ(x+z)),(x, z)∈R2, bv(x, z)=(ex−κ)(1 + eγ σ(x+z)), x ≥c(z), z ∈R, bvx(x, z) = ex(1 + eγ σ(x+z)) + γ σ(ex−κ)eγ σ(x+z), x =c(z), z ∈R, bvz(x, z) = γ σ(ex−κ)eγ σ(x+z), x =c(z), z ∈R. (6.34) In the next Proposition, upon using a suitable application of Itˆo’s Lemma, we derive a probabilistic representation of the value function bv. Its proof is postponed to Appendix B. Proposition 6.13. Recall the free boundary cof (6.19)and the function gof (6.16). For any (x, z)∈R2, the value function bvcan be written as bv(x, z) = EQ (x,z)h−Z∞ 0 e−rsg(Xs, Zs) 1 {Xs≥c(Zs)}dsi.(6.35) Denote now by G(w;m, v) := 1 √2πv2e−(w−m)2 2v2, w ∈R, m ∈R, v > 0,(6.36) the density function of a Gaussian random variable with mean mand variance v2. Then, from Proposition 6.13 we obtain the following result. Proposition 6.14. Let M:= f:R7→ R:fis nondecreasing, continuous and s.t. x∗ 0≤f(z)≤x∗ 1. Then, the free boundary cof (6.19)is the unique solution in Mto the integral equation (ec(z)−κ)(1 + eγ σ(c(z)+z)) = Z∞ 0 e−rsZR−g(w, Zs)G(w;c(z) + µ0s, σ√s) 1 {w≥c(z)}dwds,(6.37) with gas in (6.16)and Gas in (6.36). Proof. We take x=c(z) in Proposition 6.13. Employing the continuity of the value function we find (ec(z)−κ)(1 + eγ σ(c(z)+z)) = EQh−Z∞ 0 e−rsg(Xc(z) s, Zz s) 1 {Xc(z) s≥c(Zz s)}dsi, z ∈R.(6.38) By noticing that Zzis deterministic and Xc(z) sis Gaussian under Qwith mean c(z)+µ0sand variance σ2s, we can reformulate (6.38) as (6.37), upon using (6.36). To show uniqueness one can employ a four-step-approach exploiting the superharmonic characterization of bv, as originally developed in Th. 3.1 of Peskir [51]. Since the present setting does not exhibit additional challenges, we omit details for the sake of brevity. Remark 6.15. As it turns out, the integral equation (6.37)allows to derive an integral equation for the boundary b−1of (6.10)as well. Indeed, taking z=c−1(x)in (6.37)and using (6.12)yields (ex−κ)(1 + b−1(x)) = EQh−Z∞ 0 e−rsgXx s,σ γln(Φb−1(x) s)−Xx s 1 {Φb−1(x) s≤b−1(Xx s)}dsi, x ∈R.
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 23 In particular, it follows from the latter b−1(x) = 1 ex−κEQh−Z∞ 0 e−rsgXx s,σ γln(Φb−1(x) s)−Xx s 1 {Φb−1(x) s≤b−1(Xx s)}dsi−1, x ∈R. (6.39) Notice that the domain of b−1is given by the interval [x∗ 0, x∗ 1](cf. Lemma 5.3) and hence, we do not encounter any problems when dividing by ex−κsince Assumption 6.3 guarantees ex−κ > 0for x≥x∗ 0. 7. Solution of the Optimal Execution Problem In this section, we finally return to the optimal execution problem of Section 4and provide its solution. Before we do so, it is helpful to transform the singular stochastic control problem (2.9) by arguing as for the optimal stopping problem in Sections 5and 6, respectively. Since the arguments are in the same spirit of those developed in Section 5, details are omitted (see also Section 4 in Federico et al. [30]). First, we make a change of measure as in Section 5, and for Qas introduced therein, we let dXξ t=µ0dt +σdBt−αdξt, Xξ 0−=x,(7.1) denote the dynamics of the controlled process Xξunder Q. Hence, conditionally to Xξ 0−=x, Y ξ 0−=y and Φ0=ϕ, we introduce the transformed optimal control problem V(x, y, ϕ) := sup ξ∈A(y) EQ (x,y,ϕ)Z∞ 0 e−rteXξ t−κ(1 + Φt)◦dξt,(x, y, ϕ)∈R×(0,∞)×(0,∞), (7.2) and observe that V(x, y, ϕ) = (1 + ϕ)V(x, y, ϕ 1+ϕ). Furthermore, we set Zξ t:= σ γlog(Φt)−Xξ t, z := σ γlog(ϕ)−x,(7.3) for any (x, ϕ)∈R×(0,∞), which, through an application of Itˆo-Meyer’s formula, is easily shown to have dynamics dZξ t=−1 2(µ0+µ1)dt +αdξt, Zξ 0−=z.(7.4) Finally, analogously to (6.5), we define b V(x, y, z) := V(x, y, eγ σ(x+z)) = sup ξ∈A(y) EQ (x,y,z)Z∞ 0 e−rteXξ t−κ1 + eγ σ(Xξ t+Zξ t)◦dξt,(7.5) for (x, y, z)∈ O := R×(0,∞)×R, where EQ (x,y,z)denotes the expectation conditional on Xξ 0−= x, Y ξ 0−=yand Zξ 0−=z. In the following, we introduce a candidate for the value function Vof (2.9) and - through the explicit relationships between the value functions v, v and bvalso for the value functions Vand b Vof (7.2) and (7.5). To this end, we set U(x, y, π) := 1 αZx x−αy v(x0, π)dx0,(7.6) where vdenotes the value function of (4.3). Upon using the explicit relationship (5.7) of vand vit follows that U(x, y, ϕ) := (1 + ϕ)Ux, y, ϕ 1 + ϕ= (1 + ϕ)1 αZx x−αy vx0,ϕ 1 + ϕdx0=1 αZx x−αy v(x0, ϕ)dx0,(7.7)
30 DAMMANN AND FERRARI selling the shares at a higher price. We can interpret a larger volatility coefficient as a higher level of uncertainty and thus larger price fluctuations on the market. The investor exploits the latter fact and delays selling the shares by waiting for higher prices to evolve. Figure 2. The optimal execution boundaries b(ϕ) and a(π) as well as the precommitted strategies for different values of µ1and following parameters: r= 0.07, µ0=−0.01, σ = 0.17, κ = 3, π = 0.6. In Figure 2we can observe the sensitivity of the optimal execution boundaries with respect to one of the possible drift values. Since an increase in µ1implies higher expected prices on the market, the investor again delays selling shares and waits for larger prices to evolve. Differently to a change in σ, as seen in Figure 1, the lower bound x∗ 0remains untouched by a change in µ1, since it results from the case of full information when µ=µ0. Consequently, for a strong belief towards the drift value µ0, the investor does not significantly changes her execution strategy. Figure 3. The optimal execution boundaries b(ϕ) and a(π) as well as the precommitted strategies for different values of rand following parameters: µ0= −0.01, µ1= 0.007, σ = 0.17, κ = 3., π = 0.6. Finally, Figure 3shows the effect on the boundaries aand bfor a change in r, the latter can be interpreted as the impatience of the investor. Consequently, we observe that the boundary decreases for larger values of rand hence, the investor is willing to accept a lower execution price when selling the shares for every belief.
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 31 Appendix A. Proof of Proposition 6.12 The proof follows the lines of Section 4 in [21], suitably adapted to the present setting, and it is obtained through a series of intermediate results. Let (x, z)∈R2be given and fixed and set σ∗:= σ∗(x, z) := inf{t≥0 : (Xx t, Zz t)∈ S3},bσ∗:= bσ∗(x, z) := inf{t≥0 : (Xx t, Zz t)∈int(S3)}, (A.1) and observe that σ∗=τ∗Q-a.s. on R2\∂C3due to the continuity of paths. It is crucial to show that this equality also holds for the boundary points (x0, z0)∈∂C3. As it turns out, the cases i) µ0+µ1≥0 and ii) µ0+µ1<0 should be treated in different fashions and the latter case exhibits some more technical difficulties than the first case. Let us start with case i), in which the needed result follows upon using the law of iterated logarithm. Proposition A.1. Assume that µ0+µ1≥0. Let (xn, zn)∈ C3be a sequence with (xn, zn)→ (x0, z0)∈∂C3, such that x0=c(z0). We then have τ∗(xn, zn)↓0as well as bσ∗(xn, zn)↓0Q-a.s. Proof. Fix ω∈Ω and assume that lim supn→∞ τ∗(xn, zn)(ω) =: δ > 0. Hence, there exists a subsequence (still labelled by (xn, zn)) such that Xxn t(ω)< c(Zzn t)∀n∈N,∀t∈[0, δ/2],(A.2) which is equivalent to xn+µ0t+σBt(ω)< c(zn−1 2(µ0+µ1)t)∀n∈N,∀t∈[0, δ/2]. Upon using that z7→ c(z) is continuous, we let n→ ∞ and obtain σBt(ω)≤c(z0−1 2(µ0+µ1)t)−x0−µ0t≤c(z0)−x0−µ0t=−µ0t∀t∈[0, δ/2],(A.3) where the last inequality follows from µ0+µ1≥0 and Proposition 6.5. On the other hand, by the law of iterated logarithm, there exists a sequence (tn)↓0 for all > 0 such that Btn≥(1 −)r2tnlog log 1 tn ∀n∈N.(A.4) Combining (A.3) and (A.4) implies 1 tσ(1 −)r2tlog log 1 t≤ −µ0, but since p2tlog(log(1/t))/t → ∞ for t↓0, (A.2) can only happen on a Q-null set. Thus τ∗(xn, zn)↓ 0 and by replacing the strict inequality in (A.2) by ”≤”, we obtain that bσ∗(xn, zn)↓0 as well. Notice that the proof of Proposition A.1 cannot be replicated for the case ii), in which µ0+µ1<0, since the last inequality in (A.3) does not longer apply. As is turns out, in order to prove the same result for case ii), we have to take a longer route. The reason for this lies in the fact that the process (X, Z) is moving towards the right in the state space and hence - keeping in mind that the continuation region C3of (6.18) lies below the increasing boundary ccould possibly evade from the stopping set. In the following, we show that this is not the case by adapting the procedure in of Section 4 in De Angelis [21]. As a first step, we state the following Lemma, whose proof follows the lines of Cox and Peskir [16], Corollary 8, and is thus omitted for the sake of brevity. Lemma A.2. Assume that µ0+µ1<0and r > γ 2σ|µ0+µ1|. We have Q(σ∗=bσ∗) = 1. In the next step, we aim at proving regularity of the boundary points for the stopping set S3in the sense of diffusions, that is, for (x, z)∈∂C3we have Qx,z(σ∗>0) = 0.(A.5)
32 DAMMANN AND FERRARI It is clear from Blumenthal’s 0-1 law that if (A.5) does not hold, we have Qx,z(σ∗>0) = 1. Due to the mentioned structure of the problem this is not a straightforward task, since we cannot apply an argument similar to the one on Proposition A.1. Instead, we establish the result in two steps and begin by showing that the classical smooth-fit property holds at the free-boundary, i.e. continuity of bvx(·, z). Lemma A.3. Assume that µ0+µ1<0and r > γ 2σ|µ0+µ1|. For bvof (6.5)we have bvx(·, z)∈C(R), or, equivalently, bux(·, z)∈C(R)for buof (6.15). Proof. From (6.25) we obtain 1 2σ2buxx(x, z) = rbu(x, z)−µ0bux(x, z) + 1 2(µ0+µ1)buz(x, z)−g(x, z), for (x, z)∈ C3, and due to (5.12) (which implies an analogous result for bv) we deduce that for a bounded set B, we must have that buxx is bounded on the closure of B∩C3. Moreover, we recall that bux≤0 in C3, as verified in the proof of Proposition 6.7. Aiming for a contradiction we now assume that for (x0, z0)∈∂C3, such that x0=c(z0), we have bux(x0−, z0)<−δ0,(A.6) for some δ0>0. We now take a bounded rectangular neighbourhood of (x0, z0) and let τB:= inf{t > 0 : (Xt, Zt)/∈B}. Notice that bu(x0, z0)≥EQ (x0,z0)he−r(τB∧t)bu(XτB∧t, ZτB∧t) + ZτB∧t 0 e−rsg(Xs, Zs)dsi,(A.7) from the supermartingale property of (e−rtbv(Xt, Zt))t. Recall Lemma 6.10 and since t7→ ZτB∧t is increasing, we have bu(Xx0 τB∧t, Zz0 τB∧t)≥bu(Xx0 τB∧t, z0)Q-a.s. Moreover, since the integrand on the right-hand side of (A.7) is bounded on B, we obtain bu(x0, z0)≥EQ (x0,z0)he−r(τB∧t)bu(XτB∧t, z0)−cB(τB∧t)i,(A.8) where cBis a constant depending on B. Due to the previously discussed local boundedness of buxx, we can apply Itˆo-Tanaka’s formula to the first term in the expectation of (A.8). Let LX:= 1 2σ2∂xx+µ0∂x and denote the local time of Xat x0by Lx0. Moreover, noticing that buxx(·, z0) = 0 for x > x0, we obtain EQ (x0,z0)he−r(τB∧t)bu(XτB∧t, z0)i=bu(x0, z0) + EQ (x0,z0)hZτB∧t 0 e−rs(LX−r)bu(Xs, z0) 1 {Xs6=x0}dsi −EQ (x0,z0)hZτB∧t 0 e−rsbux(x0−, z0)dLx0 si, and, combining this with (A.8), as well as noticing that (LX−r)bu(Xs, Zs) is bounded on B, we find 0≥EQ (x0,z0)hZτB∧t 0 e−rs(LX−r)bu(Xs, z0) 1 {Xs6=x0}ds −cB(τB∧t)i −EQ (x0,z0)hZτB∧t 0 e−rsbux(x0−, z0)dLx0 si ≥δ0e−rtEQ (x0,z0)[Lx0 τB∧t]−cBEQ (x0,z0)[τB∧t], where we used our assumption (A.6) in the last inequality. Since this is equivalent to cBEQ (x0,z0)[τB∧ t]≥δ0e−rtEQ (x0,z0)[Lx0 τB∧t], and EQ (x0,z0)[τB∧t]≈twhile EQ (x0,z0)[Lx0 τB∧t]≈√t(see, e.g., Peskir [52], Lemma 15), we obtain the desired contradiction. Hence, bux(·, z)∈C(R). We can now state the regularity of the boundary points.
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 33 Proposition A.4. Assume that µ0+µ1<0and r > γ 2σ|µ0+µ1|. All points (x, z)∈∂C3are regular, i.e. we have Qx,z(σ∗>0) = 0. Proof. We argue by contradiction and show that if Q(x0,z0)(σ∗>0) = 1 for some boundary point (x0, z0)∈∂C3it follows that bux(x0−, z0)<0, which contradicts Lemma A.3. As a first step, we establish an upper bound for bux. Fix (x, z)∈ C3such that x > ˜x, with the latter given by (6.14). Define τ:= τ(x) := inf{t≥0 : Xx t= ˜x+}and observe that - by strong Markov property - we have bu(x, z) = sup τ EQ (x,z)he−rτbu(˜x+, Zτ) 1 {τ>τ}+Zτ∧τ 0 e−rtg(Xt, Zt)dti.(A.9) Moreover, we let ˜τ:= ˜τ(x) := inf{t > 0 : Xx t= ˜x}, and for τ0:= τ∗(x, z) we obtain bu(x−, z) = EQ (x−,z)he−r˜τ(x−)bu(˜x, Z˜τ(x−)) 1 {τ0>˜τ(x−)}+Zτ0∧˜τ(x−) 0 e−rtg(Xt, Zt)dti.(A.10) Notice that τ(x) = ˜τ(x−). Hence, subtracting (A.10) from (A.9) yields bu(x, z)−bu(x−, z) = EQhe−rτbu(˜x+, Zz τ)−bu(˜x, Zz τ) 1 {τ0>τ}i +EQhZτ∧τ0 0 e−rtg(Xx t, Zz t)−g(Xx− t, Zz t)dti. Since (˜x+, Zz τ)∈ C3on {τ0> τ}and bux≤0 in C3(see Proposition 6.7), we must have bu(˜x, Zz τ)≥bu(˜x+, Zz τ), and we obtain bu(x, z)−bu(x−, z)≤EQhZτ∧τ0 0 e−rtg(Xx t, Zz t)−g(Xx− t, Zz t)dti. If we now divide by > 0 and let ↓0, we obtain (since τ↓˜τand τ0=τ∗(x, z)) bux(x, z)≤EQhZ˜τ∧τ0 0 e−rtgx(Xx t, Zt)dti. In the next step, we assume by contradiction that there exists (x0, z0)∈∂C3with Qx0,z0(σ∗>0) = 1 and take an increasing sequence xn↑x0such that xn>˜xfor all n∈N, which is possible due to Assumption 6.3. Let τn:= τ∗(xn, zn) and notice that τn=σn:= σ∗(xn, z0) for all n∈Ndue to continuity of paths. Furthermore, σndecreases in nand σn≥σ∗:= σ∗(x0, z0), since x7→ Xx tis increasing. Set ˜τn:= ˜τ(xn) and notice that ˜τn↑˜τ. Moreover, we let σ∞:= limn→∞ σnand have σ∞∧˜τ= lim n→∞(σn∧˜τn)≥σ∗∧˜τQ-a.s. We then obtain bux(x0−, z0) = lim n→∞ bux(xn, z0)≤lim n→∞ EQhZ˜τ∧σn 0 e−rtgx(Xxn t, Zz0 t)dti =EQhZ˜τ∧σ∞ 0 e−rtgx(Xx0 t, Zz0 t)dti<0, where we used x0>˜xas well as ˜τ∧σ∞>0 due to our assumption Qx0,z0(σ∞≥σ∗>0) = 1. But this contradicts Lemma A.3 and the claim follows. As a corollary of Lemma A.2 and Proposition A.4 we obtain Corollary A.5. Assume that µ0+µ1<0and r > γ 2σ|µ0+µ1|. Then, for all (x, z)∈R2we have Qx,z(τ∗=σ∗=bσ∗)=1.
34 DAMMANN AND FERRARI This result allows us to state the continuity result of the optimal stopping time with respect to the initial data. Lemma A.6. Assume that µ0+µ1<0and r > γ 2σ|µ0+µ1|. We have limn→∞ τ∗(xn, zn) = τ∗(x, z) for any (x, z)∈R2and any sequence (xn, zn)→(x, z). In particular, if (x, z)∈∂C3, the limit is zero. Proof. Let (x, z)∈R2and denote τn:= τ∗(xn, zn) as well as τ:= τ∗(x, z) for simplicity. In order to show lower-semicontinuity, we fix ω∈Ω ouside of a null-set. For τ(ω) = 0 we are finished and thus assume τ(ω)> δ > 0. Due to Proposition 6.11 there exists kδ,ω >0 such that c(Zt(ω)) −Xt(ω)> kδ,ω, for all t∈[0, δ]. The map (t, x, z)7→ c(Zz t(ω)) −Xx t(ω) is uniformly continuous on any compact [0, δ]×K, hence we can find Nω≥1 such that for all n≥Nωand t∈[0, δ] c(Zzn t(ω)) −Xxn t(ω)> kδ,ω, and therefore lim infnτn(ω)≥δ. Since ωand δwere arbitrary, we obtain lim infnτn≥τQ-a.s. and thus lower-semicontinuity. By employing similar arguments we can show lim supnbσn≤bσQ-a.s. and the claim thus follows together with Corollary A.5. Before we finally state the proof of Proposition 6.12, we can derive a probabilistic representation of vxby employing arguments similar to those employed in the proof of Lemma 6.6. Lemma A.7. For all (x, z)∈R2\∂C3, we have bvx(x, z) = EQ (x,z)he−rτ∗eXτ∗(1 + eγ σ(Xτ∗+Zτ∗)) + γ σ(eXτ∗−κ)eγ σ(Xτ∗+Zτ∗)) 1 {τ∗<∞}i. We are therefore ready to prove Proposition 6.12. Proof of Proposition 6.12.The first statement trivially holds true for (x, z)∈int(S3) and (x, z)∈ C3, due to the result in Lemma 6.2. It thus remains to prove that Ox,zbvis continuous across the boundary ∂C3. Let (x0, z0)∈∂C3and take a sequence (xn, zn)→(x0, z0) with τn:= τ∗(xn, zn). For a fixed t > 0, we notice (Xt, Zt)∈ C3on {τn> t}and thus, upon using tower and Markov property, we obtain bvx(xn, zn) = EQ (xn,zn)he−rτneXτn(1 + eγ σ(Xτn+Zτn)) + γ σ(eXτn−κ)eγ σ(Xτn+Zτn) 1 {τn≤t}i +EQ (xn,zn)he−rtbvx(Xt, Zt) 1 {τn>t}i. Due to Assumption 4.1 we can invoke dominated convergence as well as Lemma A.6 to obtain lim n→∞ bvx(xn, zn) = ex0(1 + eγ σ(x0+z0)) + γ σ(ex0−κ)eγ σ(x0+z0)=∂ ∂x(ex−κ)(1 + eγ σ(x+z))(x0,z0), and hence, the continuity of bvxacross the optimal boundary. The continuity of bvzacross the free boundary follows similarly. For the last claim we observe that Lemma 6.2 implies 1 2σ2bvxx(x, z) = rbv(x, z)−µ0bvx(x, z) + 1 2(µ0+µ1)bvz(x, z),(A.11) for all (x, z)∈ C3. But the right-hand side of (A.11) only involves functions which are continuous on R2, hence we deduce that bvxx admits a continuous extension on C3and is therefore bounded therein. It follows that bvx(·, z) is locally Lipschitz continuous on C3, with a Lipschitz constant K(z) that is locally bounded on R. Now, because bvx(·, z) is infinitely many times continuously differentiable in the stopping region S3(and hence locally bounded therein as well), we conclude that bvxx ∈L∞ loc(R2).
OPTIMAL EXECUTION UNDER PARTIAL OBSERVATION 35 Appendix B. Proof of Proposition 6.13 Proof. Let R > 0 and define τR:= inf{t≥0 : |Xt| ≥ Ror |Zt| ≥ R}. Since bv∈C1(R2) and bvxx ∈L∞ loc(R2), we can apply a weak version of Ito’s Lemma (see, e.g., Bensoussan and Lions [7], Lemma 8.1 and Th. 8.5, pp. 183-186) up to the stopping time τR∧Tfor some T > 0, which results in bv(x, z) = EQ (x,z)he−r(τR∧T)bv(XτR∧T, ZτR∧T)−ZτR∧T 0 e−rs(LX,Z −r)bv(Xs, Zs)dsi.(B.1) The right-hand-side of (B.1) is well-defined, because Zis deterministic, Xhas an absolutely continuous transition density and LX,Z bvis defined up to a set of zero Lebesgue measure. Since bvsolves the free-boundary problem (6.34), we have (LX,Z −r)bv(x, z)=(LX,Z −r)bv(x, z) 1 {x<c(z)}+ (LX,Z −r)bv(x, z) 1 {x≥c(z)}=g(x, z) 1 {x≥c(z)}, for almost all (x, z)∈R2. Using again that the transition density of Xis absolutely continuous with respect to the Lebesgue measure, equation (B.1) becomes bv(x, z) = EQ (x,z)he−r(τR∧T)bv(XτR∧T, ZτR∧T)−ZτR∧T 0 e−rsg(Xs, Zs) 1 {x≥c(z)}dsi. Now, upon employing a change of measure as in Section 5, we obtain EQ (x,z)e−r(τR∧T)|bv(XτR∧T, ZτR∧T)|=EQ (x,z)e−r(τR∧T)|v(XτR∧T, eγ σ(XτR∧T+ZτR∧T))| ≤K1EQ (x,exp( γ σ(x+z))e−r(τR∧T)eXτR∧T(1 + ΦτR∧T) =K1(1 + eγ σ(x+z))E(x,π)e−r(τR∧T)eXτR∧T,(B.2) where π=eγ σ(x+z)/(1 + eγ σ(x+z)). Due to Assumption 4.1, it is easy to verify that taking limits in (B.2) yields lim T↑∞ lim R↑∞ EQ (x,z)he−r(τR∧T)bv(XτR∧T, ZτR∧T)i= 0.(B.3) Furthermore, EQ (x,z)hZτR∧T 0 e−rsg(Xs, Zs) 1 {x≥c(z)}dsi≤EQ (x,z)hZ∞ 0 e−rs|g(Xs, Zs)|dsi ≤EQ (x,exp( γ σ(x+z))hZ∞ 0 e−rseXs(r−1 2σ2−µ0) + rk + ΦseXs(r−1 2σ2−µ1) + rkdsi ≤EQ (x,exp( γ σ(x+z)hZ∞ 0 e−rseXs(r−1 2σ2−µ0) + rkdsi + (1 + eγ σ(x+z))E(x,π)hZ∞ 0 e−rseXs(r−1 2σ2−µ1) + rkdsi<∞,(B.4) where π=eγ σ(x+z)/(1 + eγ σ(x+z)) and the last inequality follows again from Assumption 4.1. Hence, given the finiteness of the expectation in (B.4), we can apply dominated convergence theorem in order to interchange expectation and limits as R↑ ∞ and T↑ ∞. Combining this result with (B.3) gives (6.35), which completes our proof. Funding The authors gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - SFB 1283/2 2021 - 317210226.
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