Applied Mathematics & Optimization (2022) 86:25
https://doi.org/10.1007/s00245-022-09885-w
What if We Knew What the Future Brings? Optimal
Investment for a Frontrunner with Price Impact
Peter Bank1·Yan Dolinsky2·Miklós Rásonyi3
Accepted: 1 May 2022 / Published online: 6 July 2022
© The Author(s) 2022
Abstract
In this paper we study optimal investment when the investor can peek some time units
into the future, but cannot fully take advantage of this knowledge because of quadratic
transaction costs. In the Bachelier setting with exponential utility, we give an explicit
solution to this control problem with intrinsically infinite-dimensional memory. This
is made possible by solving the dual problem where we make use of the theory of
Gaussian Volterra integral equations.
Keywords Gaussian Volterra integral equation ·Inside information ·Price impact ·
Exponential utility ·Optimal investment
Mathematics Subject Classification 91G10 ·91B16
1 Introduction
Optimal investment is a tremendously rich source of mathematical challenges in
stochastic control theory. The key driver in this problem is the tradeoff between risk
P. Bank is supported in part by the GIF Grant 1489-304.6/2019. Y. Dolinsky is supported in part by the
GIF Grant 1489-304.6/2019 and the ISF Grant 230/21. M. Rásonyi thanks for the support of the
“Lendület” Grant LP 2015-6 of the Hungarian Academy of Sciences.
BPeter Bank
Yan Dolinsky
Miklós Rásonyi
rasonyi@renyi.hu
1Department of Mathematics, TU, Berlin, Germany
2Department of Statistics, Hebrew University of Jerusalem, Jerusalem, Israel
3Alfréd Rényi Institute of Mathematics and Eötvös Loránd University, Budapest, Hungary
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25 Page 2 of 24 Applied Mathematics & Optimization (2022) 86 :25
and return. Thus, information on the investment opportunities is playing a role which
is as important for the mathematical theory as it is in practice where investors go at
great lengths to secure even the slightest advantage in knowledge. So it is no wonder
that insider information has been widely studied in the literature; see, for instance, [2,
3,5,20,24] where an investor obtains extra information about the stock price evolu-
tion at some fixed point in time. By contrast to these studies, the present paper takes a
more dynamic view on information gathering and affords the investor the opportunity
to continually peek units of time into the future. Closest to such an investor in
reality may be high-frequency traders (“frontrunners”) that get access to order flow
information earlier or are able process it faster than their competition. To the best of
our knowledge, this paper is the first continuous-time stochastic control paper with
such a feature, apart from the optimal stopping problem of [9].
Of course, perfect knowledge about future stock prices easily lets optimal invest-
ment problems degenerate and so it is of great interest to understand how market
mechanisms may curb an investor’s ability to take advantage of this extra informa-
tion. A most satisfactory approach from an economic point of view is the equilibrium
approach due to Kyle [22] where the insider knows the terminal stock price right from
the start and internalizes the impact of her orders on market prices. Generalizations of
this approach are challenging; see [6,7,28] and the references therein. For dynamic
information advantages in this context, we refer to [11,12] who consider an insider
receiving a dynamic signal on, respectively, the terminal asset price or the traded firm’s
default time. These models, however, do not get close to addressing the intrinsically
infinite-dimensional information structure of our peek-ahead setting. Fortunately, also
the much simpler market impact model of [1] that just imposes quadratic transaction
costs for the investor turns out to be sufficient friction to make the optimal investment
problem viable. In an insider model where additional information is obtained just once,
[4] use such a friction for optimal portfolio liquidation. A combination between Kyle’s
equilibrium setting and quadratic price impact costs is solved in [8]. With peek-ahead
information as in the present paper, [14] study super–replication, albeit in a discretized
version of the Bachelier model.
It is in the continuous-time Bachelier model that the present paper provides its
main result, namely the explicit optimal investment strategy for an exponential utility
maximizer who knows about future prices time units before they materialize in
the market, but cannot freely take advantage of her extra knowledge due to quadratic
transaction costs. The optimal policy turns out to be a combination of two trading
incentives. On the one hand, there is the urge to trade towards the optimal frictionless
position given by the well-known Merton ratio. On the other hand, there is the desire
to take advantage of the next stock price moves and this contributes to the optimal
turnover rate through an explicitly given average of stock prices over the window of
length on which our investor has extra information.
Due to its peek-ahead feature, our optimal control problem can be viewed as a
contribution to pathwise stochastic control. A closely related work is [10] where the
authors studied a hidden stochastic volatility model with a controller who has full
information on the extra noise. The theory of delayed or partial information also
shares the infinite-dimensional pathwise control issues we need to address here; see
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Applied Mathematics & Optimization (2022) 86 :25 Page 3 of 24 25
the recent papers [25,26] and the references therein. Finally, our control theoretic
setting is also related to models discussed in the monograph [15].
Instead of dynamic programming (which would be challenging in this infinite mem-
ory setting; cf. [15]), our methodology is based on duality. For the case of exponential
utility and quadratic transaction costs, this theory is developed with flexible infor-
mation flow in great generality in an essentially self-contained appendix. It shows
that the primal optimal control is determined by the conditional expectation of the
terminal stock price under the dual optimal probability measure. For the Brownian
framework that we focus on in the main body of the paper, we derive a particularly
convenient representation of the dual target functional which leads to deterministic
variational problems. These problems can be solved explicitly, and results from the
theory of Gaussian Volterra integral equations [18,19] allow us both to construct the
solution to the dual problem and to compute the primal optimal strategy. These Gaus-
sian Volterra integral equations also occur in [13] albeit in the rather different context
of (no) arbitrage criteria in fractionally perturbed financial models.
In Section 2we specify our model and formulate and interpret our main result.
Section 3contains the proof of the main result and the appendix A presents the duality
results necessary for these developments.
2 Problem Formulation and Main Result
We consider an investor who knows about market movements some time before they
happen, but cannot arbitrarily exploit them due to market frictions. Specifically, apart
from a riskless savings account bearing zero interest (for simplicity), the investor has
the opportunity to trade in a risky asset with Bachelier price dynamics
St=s0+μt+σWt,t≥0,
where s0∈Ris the initial asset price, μ∈Ris the constant drift, σ>0 is the constant
volatility and Wis a one-dimensional Brownian motion supported on a complete
probability space (, F,P). Rather than having access to just the natural augmented
filtration (FS
t)t≥0for making investment decisions, we assume that our investor can
peek ∈[0,∞)time units into the future, and so her information flow is given by
the filtration
G
t:= FS
t+,t≥0.
Remark 2.1 As suggested by an anonymous referee, one could more generally consider
a non-decreasing time shift τ:[0,∞)→[0,∞)with τ(t)≥tto model time-varying
ability to peek ahead. To keep the exposition here as simple as possible, we leave this
extension of our model as a topic for future research.
Taking advantage of the inside information is impeded by the investor’s adverse market
impact. Following [1], we model this impact in a temporary linear form and, thus, when
at time tthe investor turns over her position tat the rate φt=˙
tthe execution price
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is St+
2φtfor some constant >0. As a result, the profits and losses from trading
are given by
V0,φ
T:= 0(ST−S0)+T
0
φt(ST−St)dt −
2T
0
φ2
tdt,(2.1)
where, for convenience, we assume that the investor marks to market her position
T=0+T
0φtdt in the risky asset that she has acquired by time T>0.
Fixing a time horizon T>0, the natural class of admissible strategies is then
A:= φ=(φt)t∈[0,T]:φis G-optional with T
0
φ2
tdt <∞a.s..
The investor’s preferences are described by an exponential utility function
u(x):= −exp(−αx), x∈R,
with constant absolute risk aversion parameter α>0, and her goal is thus to
Maximize Eu(V0,φ
T)=E−exp −αV0,φ
Tover φ∈A.(2.2)
The paper’s main result is the following solution to this optimization problem:
Theorem 2.2 In the utility maximization problem (2.2), the investor’s optimal turnover
rate ˆ
φtat time t ∈[0,T]depends on the risk-liquidity ratio
ρ:= ασ2
,
on the position ˆ
t=0+t
0ˆ
φsds acquired so far and the privileged information on
the next stock prices (St+s)s∈[0,]in the feedback form
ˆ
φt=1
¯
S
t−St+ϒ(T−t)
μ
ασ2−ˆ
t,(2.3)
where ¯
Sis the stock price average given by
¯
S
t:= 1−ϒ(T−t)S(t+)∧T+ϒ(T−t)1
0
St+sds (2.4)
with ϒ(τ) =√ρtanh(√ρ(τ−)+)/(1+√ρtanh(√ρ(τ−)+)). The maximal
utility this policy generates is
max
φ∈AE−exp −ασ V0,φ
T
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Applied Mathematics & Optimization (2022) 86 :25 Page 5 of 24 25
=−exp α√ρ
2 coth √ρT0−μ
ασ22−1
2
μ2
σ2T
·exp −1
2T
0
(s∧)ρ
1+(s∧)√ρtanh √ρ(T−s)ds.(2.5)
Our feedback description (2.3) can be interpreted as follows: First, without priv-
ileged information, i.e. for =0, we have ¯
S
t=Stand, therefore, the first term
in (2.3) vanishes leaving us with the optimal policy
ˆ
φt=√ρtanh(√ρ(T−t)) μ
ασ2−ˆ
t,t∈[0,T].
So the uninformed agent will trade towards the optimal position μ/(ασ2)well known
from the frictionless Merton problem. Due to the impact costs, she does so with finite
urgency √ρtanh(√ρ(T−t)). With a long time to go, this urgency is essentially √ρ
and thus dictated by the risk/liquidity ration ρ=ασ2/;astapproaches the time
horizon T, the urgency vanishes because, towards the end, position improvements
have an ever shorter time to yield risk premia but the investor still has to pay the same
impact costs that obtain at the start of trading. 1
For the informed agent, i.e. for >0, the desire to be close to the Merton ratio
persists, but the urgency reduces to
ϒ(T−t)
=√ρtanh(√ρ(T−t−)+)
1+√ρtanh(√ρ(T−t−)+),
leaving “some air” to take advantage of the knowledge on future price movements.
This is done by averaging out in (2.4) the latest relevant stock price available to the
investor, S(t+)∧T,with the mean stock price 1
0St+sds to be realized over the next
time units in an effort to assess the earnings potential over today’s stock price St. Put
into relation with the impact costs , this yields the second contribution (¯
S
t−St)/
to the optimal turnover rate. The weight that this assessment of earnings assigns to the
average stock prices is given by ϒ(T−t)∈[0,1]; it is about √ρ/(1+√ρ)
when there is still a lot of time to go, but vanishes completely as soon as T−t≤,
i.e. as soon as full knowledge of stock price movements over the relevant time span
[0,T]is attained. In this terminal regime also the ambition to be close to the Merton
ratio is wiped out and the investor just chases the earning potential ST−Stfrom the
stock, of course still in a tradeoff against the liquidity costs ; this latter effect is also
immediate from separate, pointwise optimization over φtin the representation (2.1)
of profits and losses (which leads to φ∗
t=(ST−St)/,t∈[0,T], an admissible
strategy as soon as STbecomes known). Figure 1illustrates the different components
important for the optimal strategy along a typical trajectory of price fluctuations.
1We will prove this result along the way to our main result with future knowledge >0. Let us note
though that, for =0, a closely related result is obtained by dynamic programming techniques in [27]
who, in contrast to our setting, impose a liquidation constraint ˆ
T=0 and assume μ=0.
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Fig. 1 The first of these illustrations shows an evolution of the stock price S(blue), the corresponding
average S(orange) along with the underlying weight ϒ; the second shows the resulting trading rates
due to “frontrunning” (grey) and due to tracking the Merton portfolio (black); the third display shows the
ensuing stock position (red) together with the Merton ratio μ/(ασ2) (light red). Parameters where chosen
as s0=0, μ=.1, σ=.3, T=10, =1, α=.03, 0=0, =.01
The monetary value of being able to peek ahead by is best described by the
certainty equivalent
c() =−1
αlog
maxφ∈AE−exp −αV0,φ
T
maxφ∈A0E−exp −αV0,φ
T
=1
2αT
0
(s∧)ρ
1+(s∧)√ρtanh √ρ(T−s)ds (2.6)
determined by comparing the utility attainable for an informed investor (with admis-
sible strategy set A) and an uninformed one (who is confined to strategies from
the smaller class A0)2. Interestingly, the certainty equivalent does not depend on the
stock’s risk premium μ, but is determined by the risk/liquidity ratio ρ=ασ2/,the
investor’s time horizon Tand the time units she can look ahead. Except for a period
of length and with a lot of time to go, it accrues at about the rate ρ /(2(1+√ρ))
which increases with to the upper bound √ρ/2, revealing again the curb frictions
put on the earning potential of even extreme information advantages.
The proof of Theorem 2.2 is carried out in the next section. It is obtained by solving
the dual problem to
Minimze
EQ0(ST−S0)+1
2T
0EQST|G
t−St
2dt+1
αEQlog dQ
dP
over Q∼Pwith finite relative entropy EQlog dQ
dP<∞.(2.7)
The corresponding duality theory holds true beyond the Brownian framework spec-
ified here and is developed in a self-contained manner in the Appendix A as a second
key contribution of our paper.
2Theintegralin(2.6) can be computed explicitly, but the resulting formulae turn out to be not more
informative than the above integral and are therefore omitted.
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Applied Mathematics & Optimization (2022) 86 :25 Page 7 of 24 25
3 Proof of Theorem 2.2
Let us first note that it suffices to treat the case
S=W, i.e., without loss of generality s0=0,σ =1,μ=0.(3.1)
Indeed, by passing from α,,μto, respectively, α=ασ,=/σ ,μ=μ/σ,the
utility with σ=1 obtained from a given strategy will coincide with the one obtained
from this strategy under the original parameters. Moreover, rewriting the expected
utility under P∼Pwith density
dˆ
P
dP|FW
T:= exp −μWT−1
2μ2T,
under which W
t=Wt+μt,t≥0, is a driftless Brownian motion, the expected
utilities under Pcoincide, up to the factor exp 1
2μ2T, with those under Pif we start
with
0=0−μ/(ασ2)rather than 0risky assets.
The proof of Theorem 2.2 will be accomplished via the dual problem whose proper-
ties are summarized in the following proposition which is an immediate consequence
of the general duality results presented in Appendix A.
Proposition 3.1 Denoting by Qthe set of all probability measures Q∼Pwith finite
entropy
EQlog dQ
dP<∞
relative to P, we have
max
φ∈A−1
αlog Eexp −αV0,φ
T
=min
Q∈QEQ0(ST−S0)+1
αlog dQ
dP+1
2T
0EQ(ST|G
t)−St
2dt.
(3.2)
Furthermore, the minimizer ˆ
Qfor the dual problem is unique and yields via
ˆ
φt:= Eˆ
QST|G
t−St
,t∈[0,T],(3.3)
the unique optimal portfolio for the primal problem.
Proof Follows from Proposition A.2 below with the choice Gt:= G
tafter noting that
St/√tis standard Gaussian and so
sup
t∈[0,T]
E[exp(aS2
t)]≤E[exp(aS2
T)]<∞,
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clearly holds for some small enough a>0.
In order to solve the utility maximization problem it therefore suffices to find the
minimizer ˆ
Qof the dual problem and work out the conditional expectation in (3.3).
This is the path we will follow for the rest of this section. In a first step we derive a
particularly convenient representation for the target functional of our dual problem:
Lemma 3.2 The dual infimum in (3.2)coincides with the one taken over all Q∈Q
whose densities take the form
dQ
dP=exp −T
0
θtdWt−1
2T
0
θ2
tdt(3.4)
for some bounded and adapted θchanging values only at finitely many deterministic
times. For such Qthe induced value (2.7)for the dual problem can be written as
EQ0(ST−S0)+1
αlog dQ
dP+1
2T
0EQST|G
t−St
2dt
=−0T
0
atdt +1
2αT
0
a2
tdt +1
2T
0T
t
audu2
dt
+T
0
EQ1
2αT
s
l2
t,sdt +1
2T
sT
t
lu,sdu2
dt
+s∧
21−T
s
lu,sdu2ds
where, for t ∈[0,T],a
tand lt,. are determined by the Itô-representations
θt=at+t
0
lt,sdWQ
s(3.5)
with respect to the Q-Brownian motion W Q
s=Ws+s
0θrdr, s ≥0.
Proof For any Q∈Qthe martingale representation property of Brownian motion
gives us a predictable θwith EQ[log(dQ/dP)]=EQ[T
0θ2
sds]/2<∞such that
the density dQ/dPtakes the form (3.4). Using this density to rewrite the dual target
functional as an expectation under P, we can follow standard density arguments to
see that the infimum over Q∈Qcan be realized by considering the Qinduced
via (3.4)bysimpleθas described in the lemma’s formulation. As a consequence,
the Itô representations of θtin (3.5) can be chosen in such a way that the resulting
(at,lt,.)are also measurable in t: in fact they only change when θchanges its value,
i.e., at finitely many deterministic times. This joint measurability will allow us below
to freely apply Fubini’s theorem.
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Applied Mathematics & Optimization (2022) 86 :25 Page 9 of 24 25
Let us rewrite the dual target functional in terms of aand l. In terms of θand the
Q-Brownian motion WQ, it reads
EQ0(ST−S0)+1
αlog dQ
dP+1
2T
0EQST|G
t−St2dt
=EQ−0T
0
θtdt +1
2αT
0
θ2
udu
+1
2T
0WQ
(t+)∧T−WQ
t−EQT
t
θudu|G
t2
dt.(3.6)
From Itô’s isometry and Fubini’s theorem we obtain
EQT
0
θ2
udu=T
0
a2
tdt +T
0T
s
EQl2
t,sdt ds.(3.7)
Again by Fubini’s theorem it follows that
EQT
t
θudu|G
t=T
t
audu +EQT
0T
t∨s
lu,sdu dWQ
s|G
t
=T
t
audu +(t+)∧T
0T
t∨s
lu,sdu dWQ
s
for any t∈[0,T], where the last equality follows from the martingale property of
stochastic integrals. Thus, another application of Itô’s isometry yields
EQWQ
(t+)∧T−WQ
t−EQT
t
θudu|G
t2
=T
t
audu2
+EQt
0T
t
lu,sdu2
ds
+EQ(t+)∧T
t1−T
s
lu,sdu2
ds.
Plugging this together with (3.7)into(3.6) and using Fubini’s theorem then provides
us with the claimed formula for our dual target value:
EQ−0(ST−S0)+1
αlog dQ
dP+1
2T
0EQST|G
t−St2dt
=−0T
0
atdt +1
2αT
0
a2
tdt +1
2T
0T
t
audu2
dt
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25 Page 10 of 24 Applied Mathematics & Optimization (2022) 86 :25
+T
0
EQ1
2αT
s
l2
t,sdt +1
2T
sT
t
lu,sdu2
dt
+s∧
21−T
s
lu,sdu2ds.
The crucial point of the above representation is that by taking the minimum separately
over aand over l.,sfor each s∈[0,T]we obtain deterministic variational problems
that can be solved explicitly (see the next Lemma 3.3)and this deterministic minimum
yields a lower-bound for the dual target value that, using some Gaussian process theory,
will ultimately be shown to actually coincide with it (see Lemma 3.4 below).
Lemma 3.3 Recall our notation ρ=ασ2/ =α/ (because σ=1;cf.(3.1)).
(i) The minimum of the functional
−0T
0
atdt +1
2αT
0
a2
tdt +1
2T
0T
t
audu2
dt
over a ∈L2([0,T],dt)is attained for ˆa0where
ˆat=αcosh(√ρ(T−t))
cosh(√ρT),t∈[0,T].(3.8)
The resulting minimum value is −ˆ
AT2
0where
ˆ
AT=√ρtanh(√ρT)/2.(3.9)
(ii) For any s ∈[0,T], the minimum of the functional
1
2αT
s
l2
tdt +1
2T
sT
t
ludu2
dt +s∧
21−T
s
ludu2
over l ∈L2([s,T],dt)is attained at
ˆ
lt,s=ρ(s∧) cosh(√ρ(T−t))
cosh(√ρ(T−s)) +√ρ(s∧) sinh(√ρ(T−s)),t∈[s,T].(3.10)
The corresponding minimum value is
ˆ
Ls=1
2
s∧
1+(s∧)√ρtanh(√ρ(T−s)).(3.11)
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Applied Mathematics & Optimization (2022) 86 :25 Page 11 of 24 25
Proof We start with (ii). The uniqueness follows from strict convexity of the functional
to be minimized over l∈L2([s,T],dt). To write this as a standard variational prob-
lem, put H(u,v):= 1
2u2+1
2αv2for u,v ∈R, reparametrize lvia g(t)=T
tludu,
t∈[s,T], and consider, for any s∈[0,T]and any ∈R, the problem to minimize
T
sH(gt,˙gt)dt over g∈C1[s,T]subject to the constraints g(s)=,g(T)=0.
The optimization problem is convex and so it has a unique solution which has to
satisfy the Euler–Lagrange equation (for details see Section 1 in [16])
d
dt
∂H
∂˙g=∂H
∂g.
Thus, the optimizer is the unique solution of the linear ODE
¨g=ρg,g(s)=, g(T)=0,
namely
g,s(t):= sinh √ρ(T−t)
sinh √ρ(T−s),t∈[s,T].
Next, observe that for the function g(t):= T
tludu,t∈[s,T]we have ˙g=−l
where ˙gis the weak derivative of g, and so,
1
2αT
s
ψ2
tdt +1
2T
sT
t
ψudu2
dt =T
s
H(gt,˙gt)dt.
Thus, from simple density arguments (needed since gis not necessarily smooth) we
obtain that
inf
l∈L2([s,T],dt)1
2αT
s
l2
tdt +1
2T
sT
t
ludu2
dt +s∧
21−T
s
ludu2
=inf
∈RT
s
H(g,s
t,˙g,s
t)dt +s∧
2(1−)2
=1
2inf
∈Rcoth √ρ(T−s)
√ρ2+(s∧)(1−)2
where the last equality follows from simple computations.
Finally, the minimum of the above quadratic pattern (in ) is attained at
∗=− (s∧)√ρ
coth √ρ(T−s)+(s∧)√ρ.
This gives (3.10)–(3.11).
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25 Page 12 of 24 Applied Mathematics & Optimization (2022) 86 :25
The proof of (i) is almost the same as the of (ii), but slightly simpler. Observe that
inf
a∈L2([0,T],dt)−0T
0
atdt +1
2αT
0
a2
tdt +1
2T
0T
t
audu2
dt
=inf
∈R−0+T
0
H(g,0
t,˙g,0
t)dt
=inf
∈R−0+coth √ρT
2√ρ 2.
The minimum of the above quadratic pattern (in ) is attained in
˜
∗=0√ρtanh √ρT.
This gives (3.8)–(3.9).
The previous two lemmas suggest a way to construct a candidate for the solution
to the dual problem: Find ˆ
Q∼Pwhose density is given by
dˆ
Q
dP=exp −T
0ˆ
θtdWt−1
2T
0ˆ
θ2
tdt(3.12)
with
ˆ
θs=ˆas0+s
0ˆ
ls,rdˆ
Wˆ
Q
r,s∈[0,T].(3.13)
For the associated Brownian motion ˆ
W=Wˆ
Q=W+.
0ˆ
θrdr this implies the
Volterra-type integral equation
Wt+t
0ˆas0ds =ˆ
Wt−t
0s
0ˆ
ls,rdˆ
Wrds,t∈[0,T].(3.14)
Integral equations of this type occur in [18,19]; see also [13]. By considering W+
.
0ˆar0dr as a Brownian motion with respect to some probability measure which is
equivalent to P, we can apply the results from Section 6.4 in [18] (in particular see
Theorem 6.3 and its proof there). We obtain that (3.14) has a unique solution given by
ˆ
Wt=Wt+0t
0ˆasds −t
0s
0ˆ
ks,rdWr+0ˆardrds
=Wt−t
0s
0ˆ
ks,rdWrds +0t
0ˆasds −t
0s
0ˆ
ks,rˆardrds(3.15)
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Applied Mathematics & Optimization (2022) 86 :25 Page 13 of 24 25
where ˆ
kis the associated resolvent kernel characterized by the equation
ˆ
kt,s+ˆ
lt,s=t
sˆ
lt,uˆ
ku,sdu,0≤s≤t≤T.(3.16)
Moreover, ˆ
Wis a Brownian motion with respect to ˆ
Qwhich is well defined by (3.12).
As our ˆ
lis separable multiplicatively, (3.16) can be reduced to a linear ODE from
which we compute the explicit solution
ˆ
kt,s=−exp t
sˆ
lu,uduˆ
lt,s,0≤s≤t≤T.(3.17)
We are now in a position to solve the dual problem:
Lemma 3.4 The dual infimum (3.2)is attained by ˆ
Q∼Pwith density
dˆ
Q
dP=exp −T
0ˆ
θtdWt−1
2T
0ˆ
θ2
tdt
for ˆ
θconstructed in (3.13)with ˆ
W as given by (3.15);this ˆ
W coincides with the ˆ
Q-
Brownian motion induced by the P-Brownian motion W via Girsanov’s theorem. The
value of the dual problem is
−2
0√ρ
2 coth √ρT+T
0
1
2
(s∧)
1+(s∧)√ρtanh √ρ(T−s)ds.(3.18)
Proof The construction of ˆ
Q,ˆ
Wand ˆ
θhas already been established by the preceding
discussion. It is readily checked that ˆ
Qhas finite entropy relative to Pand so ˆ
Q∈Q.
Note that ˆ
Wand Wgenerate the same filtration because of (3.14) and (3.15) and so we
can follow the same reasoning as in the proof of Lemma 3.2 to obtain its representation
for the dual target functional also for ˆ
Q. Recalling the minimizing properties of ˆaand
ˆ
l.,s,s∈[0,T], it then follows that ˆ
Qsolves the dual problem with value (3.18).
By (3.2) the above value (3.18) of the dual problem already yields the claimed
value (2.5) for our primal utility maximization problem. For the completion of the
proof of Theorem 2.2 it therefore remains to work out the optimal turnover policy ˆ
φ.
Due to its dual description (3.3), it suffices to compute Eˆ
QST|G
t=Eˆ
QWT|Ft+.
Recalling the Volterra-type equation (3.14) and using Fubini’s theorem we can write
WT=ˆ
WT−T
0ˆau0+u
0ˆ
lu,sdˆ
Wsdu
=T
01−T
sˆ
lu,sdudˆ
Ws−T
0ˆaudu0.
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25 Page 14 of 24 Applied Mathematics & Optimization (2022) 86 :25
Thus, for any t∈[0,T], we find
Eˆ
QST|G
t
=(t+)∧T
01−T
sˆ
lu,sdudˆ
Ws−T
0ˆaudu0
=(t+)∧T
01−T
sˆ
lu,sdudWs−s
0ˆ
ks,rdWrds
+0(t+)∧T
01−T
sˆ
lu,sduˆasds −s
0ˆ
ks,rˆardr ds−T
0ˆaudu
where in the second step we used (3.15) to get an expression in terms of the original
input to our problem Wrather than ˆ
W. The structure of this expression suggests to
consider for X=Wand X=.
0ˆasds the integral operator
IT
t(X):= (t+)∧T
01−T
sˆ
lT
u,sdudXs−s
0ˆ
kT
s,rdXrds
for continuous paths X. Notice that the dX-integrals can be defined through integration
by parts which reveals in particular that IT
t(X)depends continuously on X; notice
also that we used the notation ˆ
lTand ˆ
kTin lieu of land kto emphasize that these
kernels depend on the time horizon T. In conjunction with (3.3) and St=Wt,this
provides us with a (somewhat) explicit ‘open loop’ expression of the optimal turnover
policy:
ˆ
φt=1
IT
t(W)−Wt+0IT
t.
0ˆaT
sds−T
0ˆaT
udu,(3.19)
where, again, ˆaTis used to recall that ˆaof (3.8) depends on T.
To establish the policy’s more informative feedback description given in Theo-
rem 2.2, we note next that dynamic programming holds for our problem:
Lemma 3.5 The optimal policy ˆ
φof (3.19)can alternatively be described in the form
ˆ
φt=1
IT−t
0(Wt+.−Wt)−Wt
+ˆ
tIT−t
0.
0ˆaT−t
sds−T−t
0ˆaT−t
udu (3.20)
where ˆ
t=0+t
0ˆ
φsds for t ∈[0,T].
Proof The righthand side of (3.19) gives us for each time horizon Ta continuous
functional T:R×C[0,T]→C[0,T]such that for any initial stock position
0∈Rand any stock price evolution W,T(0,W|[0,T])is the correspondingly
optimal strategy ˆ
φfor the utility maximization problem.
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Applied Mathematics & Optimization (2022) 86 :25 Page 15 of 24 25
Assume by contradiction that the statement of our lemma does not hold. Then,
by continuity of sample paths of ˆ
φ, there exists t0∈[0,T]such that with positive
probability ˆ
φt0does not coincide with the righthand side of (3.20). Now consider the
strategy ˜
φthat coincides with ˆ
φup to time t0when it switches to
˜
φt:= T−t0ˆ
t0,W.+t0−Wt0t−t0
,t∈[t0,T].
For any strategy φ, we can write the contribution over the interval [t0,T]to the resulting
terminal wealth as
V0,φ
T−V0,φ
t0=ˆ
t0(ST−St0)+T
t0
t(ST−St)dt −
2T
t0
φ2
tdt =: Vt0,,φ
[t0,T],
where t0:= 0+t0
0φtdt. Of course, ˜
t0:= 0+t0
0˜
φtdt =ˆ
t0. So, by the
Markov property of Brownian motion and choice of ˜
φas the unique optimal policy as
of time t0, this allows us to observe that
E−exp −αV˜
t0,˜
φ
[t0,T]|G
t0≥E−exp −αVˆ
t0,ˆ
φ
[t0,T]|G
t0,
with “>” holding on {ˆ
φt0= ˜
φt0}(i.e. where (3.20) is violated) because continuity of
ˆ
φand ˜
φensures that they will differ on an open interval once they differ at all. Since
by assumption this happens with positive probability, it follows for the unconditional
expected utility from ˜
φthat
E−exp(−αV0,˜
φ
T)=Eexp(−αV0,˜
φ
t0)E−exp −V˜
t0,˜
φ
[t0,T]|G
t0
>Eexp(−αV0,˜
φ
t0)E−exp −Vˆ
t0,ˆ
φ
[t0,T]|G
t0
=E−exp(−αV0,ˆ
φ
T),
contradicting the optimality of ˆ
φ.
As a consequence of this dynamic programming result, it suffices to verify our
feedback policy description (2.3) for time t=0:
Lemma 3.6 The optimal initial turnover rate is
ˆ
φ0=1
1+√ρtanh(√ρ(T−)+)
S∧T
(3.21)
+√ρ
coth(√ρ(T−)+)+√ρ∧T
0
Ss
ds −S0
+√ρ
coth(√ρ(T−)+)+√ρμ
ασ2−0.
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25 Page 16 of 24 Applied Mathematics & Optimization (2022) 86 :25
Proof In view of (3.19), we need to compute for X=Wand X=.
0ˆaudu the
operator
IT
0(X)=∧T
01−T
sˆ
lt,sdtdXs−s
0ˆ
ks,rdXrds
=∧T
01−T
sˆ
lt,sdtdXs−I(3.22)
with
I:= ∧T
01−T
sˆ
lt,sdts
0ˆ
ks,rdXrds
=∧T
0∧T
rˆ
ks,rds −T
rt∧
rˆ
lt,sˆ
ks,rds dtdXr(3.23)
where the last equality is due to Fubini’s theorem. For t∈[r,∧T], the kernel
identity (3.16) shows that the second ds-integral in (3.23)givesˆ
kt,r+ˆ
lt,r.Fort∈
( ∧T,T], we note that <Tand we let ntdenote the numerator in the definition
of ˆ
lt,. in (3.10) to write ˆ
lt,r=ˆ
l,rnt/n. It follows by another use of the kernel
identity (3.16) that for such tthe second ds-integral above amounts to
rˆ
lt,sˆ
ks,rds =
rˆ
l,sˆ
ks,rds nt
n=ˆ
k,r+ˆ
l,rnt
n
=ˆ
lt,r−exp
rˆ
lu,uduˆ
lt,r
where we used (3.17) in the final step. Plugging all this into (3.23), we see that the
contribution to the dt-integral from [r,∧T]is partially cancelled by the first ds-
integral there, leaving us with
I=∧T
0−T
rˆ
lt,rdt +T
∧T
exp
rˆ
lu,uduˆ
lt,rdtdXr.
Inserting this into (3.22), we see a cancellation of integrals over ˆ
land arrive at
IT
0(X)=∧T
01−T
∧T
exp
sˆ
lu,uduˆ
lt,sdtdXs
=∧T
0
(1−fT(s))dXs(3.24)
where in view of (3.10)wehave
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Applied Mathematics & Optimization (2022) 86 :25 Page 17 of 24 25
fT(s):=exp ∧T
s
uρ
1+u√ρtanh(√ρ(T−u))du
·s√ρsinh(√ρ(T−)+)
cosh(√ρ(τ −s)) +s√ρsinh(√ρ(T−s))
=s√ρsinh(√ρ(T−)+)
cosh(√ρ(T−)+)+√ρsinh(√ρ(T−)+).(3.25)
Now we apply (3.24)toX=Wand X=T
.ˆaudu to rewrite the open loop descrip-
tion (3.19)of ˆ
φ0as
ˆ
φ0=W∧T
(1−fT( ∧T)) +∧T
0
Ws
f
T(s)ds
−0T
∧T
ˆau
du(1−fT( ∧T)) +∧T
0T
s
ˆau
duf
T(s)ds.
We conclude the claimed representation for the optimal policy (3.21) by insert-
ing (3.25) and
T
s
ˆau
du =√ρsinh(√ρ(T−s))
cosh(√ρT),s∈[0,T],
in the above formula for ˆ
φ0.
As a final step, we need to recall the simplifying steps from the beginning of this
chapter where we reduced everything to the case S=Wunderpinning our calculations
so far. Reversing these steps then leads to the formulae given in the present lemma
which work for the general case required in our main theorem.
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Appendix A: Duality
In this appendix we develop the duality theory for the utility maximization prob-
lem (2.2) not just for the Bachelier model discussed in the rest of the paper, but for
any càdlàg price process S=(St)t∈[0,T]on a filtered probability space (, F,P)
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25 Page 18 of 24 Applied Mathematics & Optimization (2022) 86 :25
equipped with the (completed, right-continuous) filtration (Gt)t∈[0,T]to which Sis
adapted. Expectation of a real-valued random variable Xwith respect to some proba-
bility Ron Fis denoted ER[X]where the index is dropped when Q=P. Sometimes
we also use the shorthand notation ER,t[X](resp. Et[X]) instead of ER[X|Gt](resp.
E[X|Gt]).
Define the set of admissible strategies by
A:= (φt)t∈[0,T]:T
0
φ2
tdt <∞almost surely, φis an optional process
and for each φ∈Adefine the corresponding portfolio value at time Tby
V0,φ := 0(ST−S0)+T
0
(ST−St)φtdt −
2T
0
φ2
tdt,
where >0 is a given constant characterizing the strength of price impact.
Assumption A.1 There is a>0 such that
sup
t∈[0,T]
E[exp(aS2
t)]<∞.
Proposition A.2 Let Assumption A.1 be in force. Denoting by Qthe set of all proba-
bility measures Q∼Pwith finite entropy
EQlog dQ
dP<∞
relative to P, we have
max
φ∈A−1
αlog Eexp −αV0,φ
T
=inf
Q∈QEQ0(ST−S0)+1
αlog dQ
dP+1
2T
0EQ[ST|Gt]−St
2dt.
Furthermore, there is a unique minimizer ˆ
Qfor the dual problem and the process
given by
ˆ
φt:= Eˆ
Q[ST|Gt]−St
,t∈[0,T],
is the unique optimal portfolio for the primal problem.
We will prove Proposition A.2 at the end of this section, after suitable preparations.
In the rest of this section we assume α=1, =2, 0=0 for simplicity and will
write V(φ) instead of V0,φ. The general case is only notationally more involved.
We first state the “difficult” direction of the superhedging theorem in the present
context: it provides a sufficient condition for a claim to be superhedged by a suitable
strategy.
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Applied Mathematics & Optimization (2022) 86 :25 Page 19 of 24 25
Theorem A.3 Let Q∼Pbe a probability such that
sup
t∈[0,T]
EQ[|St|]<∞.(A.1)
Let W be a real-valued random variable with EQ[|W|] <∞.If
ER[W]≤1
4ERT
0
(ER,t[ST]−St)2dt(A.2)
holds for all probabilities R Qwith bounded d R/dQthen there exists φ∈Asuch
that V (φ) ≥W almost surely.
Proof It follows along the lines of the case treated in Theorem 3.9 of [17], the inte-
grability condition (A.1) being used in the arguments corresponding to those of page
2082 there.
Note that the convex conjugate of the function u(x):= −exp(−x),x∈Ris
v(y):= sup
x∈R[u(x)−xy],y≥0.
By simple calculations v(y)=yln y−y, where the convention 0 ln 0 =0 is used.
The Fenchel inequality u(x)≤v(y)+xy trivially holds for all x∈Rand y≥0.
Let Zdenote the set of non-negative random variables ξsuch that E[ξ]=1. For
each ξ, define the probability R(ξ) Pby R(ξ)(A):= E[ξ1A].LetZe:= {ξ∈Z:
E[ξ|ln ξ|] <∞}.
In the rest of this section, Assumption A.1 will be in force. Since we will follow a
standard route, described e.g. in [21], only the main steps of the proofs will be given.
Lemma A.4 For any family Z0⊂Zwith supξ∈Z0E[ξln ξ]<∞, one has
sup
ξ∈Z0,t∈[0,T]
E[ξS2
t]<∞.
Proof Consider the conjugate Orlicz spaces corresponding to the Young functions
(x)=ex−x−1 and (x)=(1+x)ln(1+x)−x, their respective norms being
denoted by ||·||,||·||, see the Appendix of [23] for definitions. Assumption A.1
then implies supt∈[0,T]||S2
t||<∞. Using Proposition A-2-2 of [23], we get that
E[ξS2
t]≤C||ξ||||S2
t||
with some constant C. The statement follows.
Lemma A.5 The functional
(ξ) := E[ξln ξ]+1
4ER(ξ) T
0
(ER(ξ),t[ST]−St)2dt(A.3)
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25 Page 20 of 24 Applied Mathematics & Optimization (2022) 86 :25
is finite and strictly convex on the convex set Ze.
Proof Finiteness of the functional follows from Lemma A.4, convexity of the set Ze
is easy. Strict convexity of the first summand defining is easy to see; convexity of
the second part is somewhat more strenuous. The main ingredient is the following: for
any random variable X∈∩
ξ∈ZeL1(R(ξ)) and sigma-algebra Hthe mapping
ξ∈Ze→ER(ξ)[(ER(ξ)[X|H])2]
is convex. Indeed, for s∈[0,1]and ξ1,ξ
2∈Zeone should check that
ER(sξ1+(1−s)ξ2)[(ER(sξ1+(1−s)ξ2)[X|H])2]
=E(sξ1+(1−s)ξ2)E[(sξ1+(1−s)ξ2)X|H]
E[sξ1+(1−s)ξ2|H]2
=EE2[(sξ1+(1−s)ξ2)X|H]
E[sξ1+(1−s)ξ2|H]
≤EsE2[ξ1X|H]
E[ξ1|H]+E(1−s)E2[ξ2X|H]
E[ξ2|H],
where, for simplicity, we assume ξ1,ξ
2>0 almost surely, the general case being only
notationally more difficult. This is equivalent, by elementary calculations, to checking
2E[ξ1X|H]E[ξ2X|H]E[ξ1|H]E[ξ2|H]
≤E2[ξ1X|H]E2[ξ2|H]+E2[ξ2X|H]E2[ξ1|H]
which is clearly true. Now applying this observation with the choice H=Gtand
X=ST−Stfor each tthe statement follows easily.
Proposition A.6 There exists a unique minimizer ξof on Zewhich is positive almost
surely. We denote J := infξ∈Ze(ξ).
Proof Uniqueness of a minimizer is immediate from the strict convexity of .Let
ξn∈Zebe a minimizing sequence for . By the Komlós theorem, there exist Césaro-
means of a subsequence (denoted by ˜
ξn) converging almost surely to some ξ.AsZe
is convex, ˜
ξn∈Ze. Convexity of implies that
(˜
ξn)→J(A.4)
still holds. As supnE[˜
ξn|ln ˜
ξn|] <∞follows from (A.4), the de la Vallée-Poussin
criterion ensures that ˜
ξnconverges in L1, too, hence ξ∈Ze.
We claim that ER(˜
ξn),t[ST]→ER(ξ),t[ST]in probability and observe that, by
Fatou’s lemma and (A.4), this entails (ξ)≤J, establishing ξas a minimizer of
in Ze. Since ˜
ξnST→ξSTalmost surely, the claimed convergence will follow from
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Applied Mathematics & Optimization (2022) 86 :25 Page 21 of 24 25
the uniform integrability condition
sup
n
E˜
ξn|ST|1{˜
ξn|ST|}≥b→0asb↑∞.(A.5)
Forthisweestimate
E˜
ξn|ST|1{˜
ξn|ST|≥b}≤E˜
ξn|ST|21/2E˜
ξn1{˜
ξn|ST|≥b}1/2
The first of the above factors is bounded uniformly in ndue to Lemma A.4. We will
get our assertion (A.5) by showing that the second factor vanishes uniformly in nas
b↑∞. For this we estimate it further:
E˜
ξn1{˜
ξn|ST|≥b}≤E˜
ξn1{ln(˜
ξn(|ST|∨1))≥ln b}≤E˜
ξn
ln+(˜
ξn(|ST|∨1))
ln b
≤1
ln bE˜
ξnln+˜
ξn+E˜
ξnln+(|ST|∨1).
The first of these last two expectations is bounded uniformly due to (A.4), the second
because of this in conjunction with Lemma A.4, and we can conclude.
To prove positivity of ξ, let us pick a strictly positive ξ0∈Ze(for instance ξ0≡1)
and define
Fs:= (sξ0+(1−s)ξ), s∈[0,1].
By optimality of ξthe right derivative F
0+is non-negative. If we had P(ξ =0)>0
then we would reach a contradiction just like in Proposition 3.1 of [21].
Theorem A.7 There exists a strategy φ†such that V (φ†)=J−ln ξ. Moreover,
sup
φ∈A
E[−exp(−V(φ))]=E[−exp(−V(φ†))].(A.6)
Proof Notice that V(φ) ≤Q:= 1
4T
0(ST−St)2dt for all φ, and Qis R(ξ)-integrable
for all ξ∈Ze, by the arguments of Lemma A.4. Let us fix now an arbitrary φ∈A
and ξ∈Ze. For each s>0, we apply the Fenchel inequality and Fubini’s theorem,
remembering the integrability of the quantity Q:
E[−exp(−V(φ))]
≤E[sξln(sξ)]−E[sξ]+Esξ−T
0
Stφtdt −T
0
φ2
tdt +T
0
STφtdt
=E[sξln(sξ)]−s+sT
0
ER(ξ) −Stφt−φ2
t+STφtdt
≤E[sξln(sξ)]−s+sEξT
0−Stφt−φ2
t+ER(ξ),t[ST]φtdt
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25 Page 22 of 24 Applied Mathematics & Optimization (2022) 86 :25
≤sln s+sE[ξln ξ]−s+s
4EξT
0
(ER(ξ),t[ST]−St)2dt.
Optimizing in swe arrive at
E[−exp(−V(φ))]≤−exp −E[ξln ξ]−1
4EξT
0
(ER(ξ),t[ST]−St)2dt.
Choosing ξ:= ξ,
E[−exp(−V(φ))]≤−e−J.(A.8)
Now we will prove that a suitable strategy φ†attains the bound given in (A.8). Let
ξ∈Zbe such that ξ≤C0ξfor some C0>0. Since ξ|ln ξ|≤C0ξ(|ln ξ|+|ln C0|),
we have, in fact, ξ∈Ze. Consider the function
Fs:= (sξ+(1−s)ξ), s∈[0,1].
Since ξis the minimizer, Fs≥F0for s∈[0,1]so the right-hand derivative satisfies
F
0+≥0.(A.9)
To simplify notation, we will write Xt:= ST−Sthenceforth. Let us calculate the
latter derivative now. It equals
E[(ξ −ξ)ln(ξ)]+1
4T
0
EEt[ξ]2Et[ξXt]Et[(ξ −ξ)Xt]−Et[ξ−ξ]E2
t[ξXt]
E2
t[ξ]dt.
Grouping the terms inside the integral that do not contain ξwe obtain
−ER(ξ)E2
t[ξXt]
E2
t[ξ].
The rest of the integrand is
E2Et[ξXt]Et[ξXt]
Et[ξ]+−Et[ξ]E2
t[ξXt]
E2
t[ξ].
The first term of the latter expression can be rewritten and estimated by Cauchy’s
inequality:
2EEt[ξXt]Et[ξXt]Et[ξ]
Et[ξ]Et[ξ]≤EE2
t[ξXt]Et[ξ]
E2
t[ξ]+EE2
t[ξXt]Et[ξ]
E2
t[ξ].
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Applied Mathematics & Optimization (2022) 86 :25 Page 23 of 24 25
Taking all terms into consideration, condition (A.9) eventually implies
E[−ξln(ξ)]
≤E[−ξln(ξ)]−1
4ER(ξ)T
0
E2
t[ξXt]
E2
t[ξ]dt+1
4ER(ξ) T
0
E2
t[ξXt]
E2
t[ξ]dt.
We now use Theorem A.3 with the choice Q=R(ξ)and R=R(ξ). We obtain from
(A.2) and (A.3) that −ln ξ+J≤V(φ†)for some φ†∈A. Since
E[−exp(−V(φ†))]≥E[−exp(ln ξ−J)]=−e−JE[ξ]=−e−J,
φ†is indeed an optimal strategy. Note also that, by (A.8), the above inequality must
be an a.s. equality so ln ξ=J−V(φ†).
Corollary A.8 The strategy
ˆ
φt:= ER(ξ),t[ST]−St
2,t∈[0,T]
is optimal for the problem (A.6).
Proof Indeed, the arguments of the previous theorem show that, for all φ∈A,ξ∈Ze
and s>0,
E[−exp(−V(φ))]
≤sln s+sE[ξln ξ]−s+s
4EξT
0
(ER(ξ),t[ST]−St)2
=sln s+sE[ξln ξ]−s+sEξV(ˆ
φ)
≤−exp −E[ξln ξ]−1
4EξT
0
(ER(ξ),t[ST]−St)2,
with equality for φ=φ†,ξ=ξand for a suitable s=s∗. This implies that ˆ
φis an
optimal strategy.
Proof of Proposition A.2 Proposition A.6 and Theorem A.7 establish that the optimal
portfolio wealth V(φ†)for the (primal) utility maximization problem (A.6) can be
found by first finding the (dual) minimizer ξof the functional (A.3) and then taking
φ†satisfying V(φ†)=J−ln ξ. Finally, from the strict concavity of the map φ→
−exp(−V(φ)) we conclude that the strategy ˆ
φwhich is given in Corollary A.8 is the
unique optimal strategy.
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25 Page 24 of 24 Applied Mathematics & Optimization (2022) 86 :25
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