# Optimal Liquidation with High Risk Aversion in the Almgren–Chriss Model: A
Case Study

Leonid Dolinskyi  Department of Finance, National University of Kyiv-Mohyla
Academy.
e.mail<EMAIL_ADDRESS>and  Yan Dolinsky  Department of Statistics,
Hebrew University of Jerusalem.
e.mail<EMAIL_ADDRESS>
###### Abstract.

We consider the Bachelier model with linear price impact. Exponential utility
indifference prices are studied for vanilla European options in the case where
the investor is required to liquidate her position at the maturity date. Our
main result is establishing a non-trivial scaling limit for a vanishing price
impact which is inversely proportional to the risk aversion. We compute the
limit of the corresponding utility indifference prices and find explicitly a
family of portfolios which are asymptotically optimal.

YD Supported in part by the GIF Grant 1489-304.6/2019 and the ISF grant 230/21

Mathematical Subject Classification (2010):

91B16, 91G10, 60H30

Keywords:

exponential utility, linear price impact, optimal liquidation

##  1\. Introduction

In financial markets, trading moves prices against the trader: buying faster
increases execution prices, and selling faster decreases them. This aspect of
liquidity, known as market depth Black (1986) or price-impact, has received
large attention in optimal liquidation problems, see, for instance, Almgren &
Chriss (2001), Schied et al. (2009), Gatheral & Schied (2011), Bayrakatar &
Ludkovski (2014), Bank & Voß (2019), Fruth et al. (2019), and the references
therein.

In this paper we consider the problem of optimal liquidation for exponential
utility function in the Almgren–Chriss model Almgren & Chriss (2001) with
linear temporary impact for the underlying asset. We compute the asymptotic
behavior of the exponential utility indifference prices where the risk
aversion goes to infinity at a rate which is inversely proportional to the
linear price impact which goes to zero. In addition we provide a family of
asymptotically optimal hedging strategies.

The main motivation for the study of the asymptotic behaviour of utility
indifference prices is that in the presence of price impact, super–replication
is prohibitively costly, see Guasoni & Rasonyi (2015). Namely, in the presence
of price impact, even in market models such as the Bachelier model or the
Black–Scholes model (which are complete in the frictionless setup) there is no
practical way to construct a hedging strategy which eliminates all risk from a
financial position. This brings us to utility indifference pricing.

We divide the proof of our main result (Theorem 2.1) into two main steps: the
proof of the lower bound and the proof of the upper bound. In the proof of the
lower bound we apply Theorem 2.2 from Dolinsky (2022) which gives a dual
representation of the certainty equivalent for the case where the investor has
to liquidate her position. This dual representation together with the Brownian
structure allows to compute the scaling limit of the utility indifference
prices. The proof of the upper bound is done by an explicit construction of a
family of portfolios which are asymptotically optimal.

The rest of the paper is organized as follows. In the next section we
introduce the setup and formulate the main results. In Section 3 we prove the
lower bound. In Section 4 we prove the upper bound. In Section 5 we derive an
auxiliary result from the field of deterministic variational analysis.

##  2\. Preliminaries and Main Results

Let $T<\infty$ be the time horizon and let $W=(W_{t})_{t\in[0,T]}$ be a
standard one dimensional Brownian motion defined on the filtered probability
space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ where
the filtration $(\mathcal{F}_{t})_{t\in[0,T]}$ satisfies the usual assumptions
(right continuity and completeness). We consider a simple financial market
with a riskless savings account bearing zero interest (for simplicity) and
with a risky asset $S=\left(S_{t}\right)_{t\in[0,T]}$ with Bachelier price
dynamics

(2.1) $S_{t}=S_{0}+\mu t+\sigma W_{t}$

where $S_{0}\in\mathbb{R}$ is the initial position of the risky asset,
$\mu\in\mathbb{R}$ is the constant drift and $\sigma>0$ is the constant
volatility.

Following Almgren & Chriss (2001), we model the investor’s market impact, in a
temporary linear form and, thus, when at time $t$ the investor turns over her
position $\Phi_{t}$ at the rate $\dot{\Phi}_{t}:=\frac{d\Phi_{t}}{dt}$ the
execution price is $S_{t}+\frac{\Lambda}{2}\dot{\Phi}_{t}$ for some constant
$\Lambda>0$. The portfolio value at the maturity date is given by

(2.2)
$V^{\Phi}_{T}:=\int_{0}^{T}\Phi_{t}dS_{t}-\frac{\Lambda}{2}\int_{0}^{T}\dot{\Phi}^{2}_{t}dt.$

In our setup the investor has to liquidate her position. Thus, the natural
class of admissible strategies which we denote by $\mathcal{A}$ is the set all
progressively measurable processes $\Phi=(\Phi_{t})_{t\in[0,T]}$ with
differentiable trajectories such that
$\int_{0}^{T}\dot{\Phi}^{2}_{t}dt<\infty$ and $\Phi_{T}=0$ almost surely. We
assume that the initial number of shares $\Phi_{0}$ is fixed.

Consider a vanilla European option with the payoff $X=f(S_{T})$ where $f$ is
of the form

(2.3) $f(x)=\max\left(0,\Theta\left(x-K\right)\right),\ \ x\in\mathbb{R}$

for some constants $\Theta,K\in\mathbb{R}$. Observe that this form includes
call/put options.

The investor will assess the quality of a hedge by the resulting expected
utility. Assuming exponential utility with constant absolute risk aversion
$\alpha>0$, the utility indifference price and the certainty equivalent price
of the claim $X$ (see, e.g., Carmona (2009) for details on indifference
prices) do not depend on the investor’s initial wealth and, respectively, take
the well-known forms

(2.4)
$\pi(\Lambda,\alpha,\Phi_{0},X):=\frac{1}{\alpha}\log\left(\frac{\inf_{\phi\in\mathcal{A}}\mathbb{E}_{\mathbb{P}}\left[\exp\left(\alpha\left(X-V^{\Phi}_{T}\right)\right)\right]}{\inf_{\phi\in\mathcal{A}}\mathbb{E}_{\mathbb{P}}\left[\exp\left(-\alpha
V^{\Phi}_{T}\right)\right]}\right)$

and

$c(\Lambda,\alpha,\Phi_{0},X):=\frac{1}{\alpha}\log\left(\inf_{\phi\in\mathcal{A}}\mathbb{E}_{\mathbb{P}}\left[\exp\left(\alpha\left(X-V^{\Phi}_{T}\right)\right)\right]\right).$

If the risk aversion $\alpha>0$ is fixed, then by applying standard density
arguments we obtain that for $\Lambda\downarrow 0$, the above indifference
price converges to the unique price of the continuous time complete
(frictionless) market given by (2.1). A more interesting limit emerges,
however, if we re-scale the investor’s risk-aversion in the form
$\alpha:=A/\Lambda$. Hence, we fix $A>0$ and consider the case where the risk
aversion is $\alpha(\Lambda):=\frac{A}{\Lambda}$.

Before we formulate the main result we need some preparations. Introduce the
functions

(2.5)
$g(x):=\sup_{y\in\mathbb{R}}\left[f(x+y)-\frac{y^{2}}{4\sigma\sqrt{A}}\right]=\max\left(0,\Theta\left(x-K\right)+\sigma\sqrt{A}\Theta^{2}\right),\
\ x\in\mathbb{R}$

and

$u(t,x):=\mathbb{E}_{\mathbb{P}}\left[g(x+\sigma
W_{T-t})\right],\quad(t,x)\in[0,T]\times\mathbb{R}.$

The term $u(t,S_{t})$ represents the price at time $t$ of a European option
with the payoff $g(S_{T})$ in the complete market given by (2.1). It is well
known that $u\in C^{1,2}([0,T)\times\mathbb{R})$ solves the PDE

(2.6) $\frac{\partial u}{\partial
t}+\frac{\sigma^{2}}{2}\frac{\partial^{2}u}{\partial x^{2}}=0\ \ \ \
\mbox{in}\ \ [0,T)\times\mathbb{R}.$

Next, let $\Lambda>0$ and let

$\rho=\rho(\Lambda):=\frac{\sigma^{2}\alpha(\Lambda)}{\Lambda}=\frac{\sigma^{2}A}{\Lambda^{2}}$

be the risk-liquidity ratio. Consider the (random) ODE on the interval $[0,T]$

(2.7)
$\displaystyle\dot{F}_{t}=\sqrt{\rho}\left(\frac{\cosh(\sqrt{\rho}(T-t))}{2\cosh^{2}\left(\frac{\sqrt{\rho}(T-t)}{2}\right)}\frac{\partial
u}{\partial
x}(t,S_{t}-\sigma\sqrt{A}F_{t})-\tanh(\sqrt{\rho}(T-t))F_{t}\right),$
$\displaystyle\mbox{with}\ \mbox{the}\ \mbox{initial}\ \mbox{condition}\ \
F_{0}=\Phi_{0}\coth(\sqrt{\rho}T).$

From the linear growth of $g$ it follows that for any $\epsilon>0$ the
functions $\frac{\partial u}{\partial x},\frac{\partial^{2}u}{\partial x^{2}}$
are uniformly bounded in the domain $[0,T-\epsilon]\times\mathbb{R}$. In
particular $\frac{\partial u}{\partial x}$ is Lipschitz continuous with
respect to $x$ in the domain $[0,T-\epsilon]\times\mathbb{R}$. Observe that
the functions
$\frac{\cosh(\sqrt{\rho}(T-t))}{2\cosh^{2}\left(\frac{\sqrt{\rho}(T-t)}{2}\right)},\tanh(\sqrt{\rho}(T-t))$
are bounded. Hence, from the standard theory of ODE (see Walter (1998),
Chapter II, Section 6) we obtain that there exists a unique solution to (2.7)
which we denote by $F^{\Lambda}=(F^{\Lambda}_{t})_{t\in[0,T)}$ and the
solution is Lipschitz continuous, and so $\lim_{t\rightarrow
T-}F^{\Lambda}_{t}$ exists. Set $F^{\Lambda}_{T}:=\lim_{t\rightarrow
T-}F^{\Lambda}_{t}$ and define

(2.8)
$\Phi^{\Lambda}_{t}:=\tanh\left(\sqrt{\rho(\Lambda)}(T-t)\right)F^{\Lambda}_{t},\
\ t\in[0,T].$

######  Theorem 2.1.

For vanishing linear price impact $\Lambda\downarrow 0$ and re-scaled high
risk-aversion $A/\Lambda$ with $A>0$ fixed, the certainty equivalent of
$X=\max\left(0,\Theta\left(S_{T}-K\right)\right)$ has the scaling limit

(2.9) $\lim_{\Lambda\downarrow
0}c(\Lambda,A/\Lambda,\Phi_{0},X)=u\left(0,S_{0}-\sigma\sqrt{A}\Phi_{0}\right)+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}.$

Moreover, the trading strategies given by (2.8) are asymptotically optimal,
i.e.

(2.10) $\lim_{\Lambda\downarrow
0}\frac{\Lambda}{A}\log\left(\mathbb{E}_{\mathbb{P}}\left[\exp\left(\frac{A}{\Lambda}\left(X-V^{\Phi^{\Lambda}}_{T}\right)\right)\right]\right)=u\left(0,S-\sigma\sqrt{A}\Phi_{0}\right)+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}.$

From Theorem 2.1 we obtain immediately the following corollary which says that
the asymptotic value of the utility indifference prices is equal to the price
of the vanilla European option with the payoff $g(S_{T})$ and the shifted
initial stock price $S_{0}-\sigma\sqrt{A}\Phi_{0}$.

######  Corollary 2.2.

For vanishing linear price impact $\Lambda\downarrow 0$ and re-scaled high
risk-aversion $A/\Lambda$ with $A>0$ fixed, the utility indifference price of
$X$ has the scaling limit

$\lim_{\Lambda\downarrow
0}\pi(\Lambda,A/\Lambda,\Phi_{0},X)=u\left(0,S_{0}-\sigma\sqrt{A}\Phi_{0}\right).$

###### Proof.

Apply (2.8) and take $X\equiv 0$ for the denominator of (2.4). ∎

######  Remark 2.3.

In the proof of the lower bound (given in the next section) we only assume
that the payoff function $f$ is Lipschitz continuous. By a more careful
analysis we can prove that in fact there is an equality, namely (2.9) holds
true for any payoff function $X=f(S_{T})$ with a Lipschitz continuous $f$.
Unfortunately, the proof of (2.10) (given in Section 4) uses the specific
structure of the payoff given by (2.3). This together with the fact that the
most common vanilla options in real markets are of the form (2.3) led us to
assume from the beginning that the payoff is of this form.

Let us emphasize that our results can be extended to the multi–asset case with
a similar proof. In the multi asset case the volatility $\sigma$ is replaced
with a positive definite matrix and the functions $\coth$ and $\tanh$ are
viewed as matrix valued functions.

######  Remark 2.4.

Theorem 2.1 can be viewed as an extension of the main result in Dolinsky &
Moshe (2022), for the case where the investor has to liquidate her portfolio
at the maturity date. In both cases (with or without liquidation) the scaling
limit of the utility indifference prices is equal to
$\mathbb{E}\left[h\left(x+\sigma W_{T}\right)\right]$ for a modified function
$h$. In the present paper

$h(x)=\sup_{y\in\mathbb{R}}\left[f(x+y)-\frac{y^{2}}{4\sigma\sqrt{A}}\right]$

while in Dolinsky& Moshe (2022) the modified payoff is smaller and given by

$h(x)=\sup_{y\in\mathbb{R}}\left[f(x+y)-\frac{y^{2}}{2\sigma\sqrt{A}}\right].$

Next, we discuss the constructed asymptotically optimal portfolios. From (2.8)
we have

(2.11)
$\dot{\Phi}^{\Lambda}_{t}=\sqrt{\rho(\Lambda)}\left(\tanh\left(\frac{\sqrt{\rho(\Lambda)}\left(T-t\right)}{2}\right)\Upsilon^{\Lambda}_{t}-\coth\left(\sqrt{\rho(\Lambda)}\left(T-t\right)\right)\Phi^{\Lambda}_{t}\right)$

where $\rho(\Lambda):=\frac{\sigma^{2}A}{\Lambda^{2}}$ and
$\Upsilon^{\Lambda}_{t}:=\frac{\partial u}{\partial
x}(t,S_{t}-\sigma\sqrt{A}F^{\Lambda}_{t})$, $t\in[0,T)$.

Thus, we have a mean reverting structure which combines trucking the
$\Delta$–hedging strategy of a modified claim $g$ and liquidating the position
at the maturity date. As time $t$ approaches maturity the weight of the
$\Delta$–hedging trading strategy becomes smaller and the investor trading is
mainly towards liquidation. This is in contrast to the asymptotically optimal
portfolios in Dolinsky and & Moshe (2022) which are just based on trucking the
appropriate $\Delta$–hedging strategy.

##  3\. Proof of the Lower Bound

In this section we prove the following statement.

######  Proposition 3.1.

For vanishing linear price impact $\Lambda\downarrow 0$ and re-scaled high
risk-aversion $A/\Lambda$ with $A>0$ fixed, we have the following lower bound

$\lim\inf_{\Lambda\downarrow 0}c(\Lambda,A/\Lambda,\Phi_{0},X)\geq
u\left(0,S_{0}-\sigma\sqrt{A}\Phi_{0}\right)+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}.$

We start with the following Lemma.

######  Lemma 3.2.

Denote by $\Gamma$ the set of all progressively measurable processes
$\theta=(\theta_{t})_{t\in[0,T]}$ such that $\theta\in
L^{2}(dt\otimes\mathbb{P})$ and let $\mathcal{M}$ be the set of all
$\mathbb{P}$–martingales $M=(M_{t})_{t\in[0,T)}$ which are defined on the
half-open interval $[0,T)$ and satisfy
$||M||_{L^{2}(dt\otimes\mathbb{P})}:=\mathbb{E}_{\mathbb{P}}\left[\int_{0}^{T}M^{2}_{t}dt\right]<\infty$.
Then, for any $\Lambda,\alpha>0$ we have

$\displaystyle c(\Lambda,\alpha,\Phi_{0},X)$
$\displaystyle\geq\sup_{(\theta,M)\in\Gamma\times\mathcal{M}}\mathbb{E}_{\mathbb{P}}\left[f\left(S_{T}+\sigma\int_{0}^{T}\theta_{t}dt\right)-\frac{1}{2\alpha}\int_{0}^{T}\theta^{2}_{t}dt\right.$
$\displaystyle\left.-\Phi_{0}(M_{0}-S_{0})-\frac{1}{2\Lambda}\int_{0}^{T}\left|S_{0}+\mu
t+\sigma\int_{0}^{t}\theta_{s}ds-M_{t}\right|^{2}dt\right].$

###### Proof.

Denote by $\mathcal{Q}$ the set of all equivalent probability measures
$\mathbb{Q}\sim\mathbb{P}$ with finite entropy
$\mathbb{E}_{\mathbb{Q}}\left[\log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\right]<\infty$
relative to $\mathbb{P}$. For any $\mathbb{Q}\in\mathcal{Q}$ let
$\mathcal{M}^{\mathbb{Q}}$ be the set of all $\mathbb{Q}$–martingales
$M^{\mathbb{Q}}=(M^{\mathbb{Q}}_{t})_{t\in[0,T)}$ which are defined on the
half-open interval $[0,T)$ and satisfy
$||M^{\mathbb{Q}}||_{L^{2}(dt\otimes\mathbb{Q})}:=\mathbb{E}_{\mathbb{Q}}\left[\int_{0}^{T}|M^{\mathbb{Q}}_{t}|^{2}dt\right]<\infty$.

From the linear growth of $f$ it follows that
$\mathbb{E}_{\mathbb{P}}\left[e^{\alpha X}\right]<\infty$. Thus, define the
probability measure $\tilde{\mathbb{P}}$ by
$\frac{d\tilde{\mathbb{P}}}{d\mathbb{P}}:=\frac{e^{\alpha
X}}{\mathbb{E}_{\mathbb{P}}\left[e^{\alpha X}\right]}.$ The Cauchy–Schwarz
inequality yields that there exists $a>0$ such that
$\mathbb{E}_{\tilde{\mathbb{P}}}\left[\exp\left(a\sup_{0\leq t\leq
T}S^{2}_{t}\right)\right]<\infty.$ Hence, Assumption 2.1 in Dolinsky (2022)
holds true. Thus, by applying Theorem 2.2 in Dolinsky (2022) for the
probability measure $\tilde{\mathbb{P}}$ and the simple equality

$\mathbb{E}_{\mathbb{Q}}\left[\log\left(\frac{d\mathbb{Q}}{d\tilde{\mathbb{P}}}\right)\right]=\mathbb{E}_{\mathbb{Q}}\left[\log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)-\alpha
X\right]+\alpha\log\left(\mathbb{E}_{\mathbb{P}}\left[e^{\alpha
X}\right]\right)\ \ \forall\mathbb{Q}\in\mathcal{Q}$

we obtain

(3.1) $\displaystyle c(\Lambda,\alpha,\Phi_{0},X)$
$\displaystyle=\sup_{\mathbb{Q}\in\mathcal{Q}}\sup_{M^{\mathbb{Q}}\in\mathcal{M}^{\mathbb{Q}}}\mathbb{E}_{\mathbb{Q}}\left[X-\frac{1}{\alpha}\log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)-\Phi_{0}(M^{\mathbb{Q}}_{0}-S_{0})-\frac{1}{2\Lambda}\int_{0}^{T}|M^{\mathbb{Q}}_{t}-S_{t}|^{2}dt\right].$

Next, Let $C[0,T]$ be the space of continuous functions
$z:[0,T]\rightarrow\mathbb{R}$ equipped with the uniform norm
$||z||:=\sup_{0\leq t\leq T}|z_{t}|$. Denote by $\hat{\Gamma}\subset\Gamma$
the set of all continuous and bounded processes
$\theta=(\theta_{t})_{t\in[0,T]}$ of the form $\theta=\tau(W)$ where
$\tau:C[0,T]\rightarrow C[0,T]$ is Lipschitz continuous and predictable (i.e.
$\tau_{t}(x)=\tau_{t}(y)$ if $x_{[0,t]}=y_{[0,t]}$). From standard density
arguments and the Lipschitz continuity of $f$ it follows that in order to
complete the proof of the Lemma it is sufficient to show that for any
$(\theta,M)\in\hat{\Gamma}\times\mathcal{M}$ we have

$\displaystyle c(\Lambda,\alpha,\Phi_{0},X)$ (3.2)
$\displaystyle\geq\mathbb{E}_{\mathbb{P}}\left[f\left(S_{T}+\sigma\int_{0}^{T}\theta_{t}dt\right)-\frac{1}{2\alpha}\int_{0}^{T}\theta^{2}_{t}dt\right.$
$\displaystyle\left.-\Phi_{0}(M_{0}-S_{0})-\frac{1}{2\Lambda}\int_{0}^{T}\left|S_{0}+\mu
t+\sigma\int_{0}^{t}\theta_{s}ds-M_{t}\right|^{2}dt\right].$

To this end let $(\theta,M)\in\hat{\Gamma}\times\mathcal{M}$ such that
$\theta=\tau(W)$ where $\tau$ as above. Consider the stochastic differential
equation (SDE)

(3.3) $dY_{t}=dW_{t}-\tau_{t}(Y)dt,\ \ t\in[0,T]$

with the initial condition $Y_{0}=0$. Theorem 2.1 from Chapter IX in Revuz &
Yor (1999) yields that there exists a unique strong solution to the above SDE.
From the Girsanov theorem it follows that there exists a probability measure
$\mathbb{Q}\in\mathcal{Q}$ such that $W^{\mathbb{Q}}:=Y$ is a Brownian motion
with respect to $\mathbb{Q}$.

From (2.1) and (3.3) we obtain that the distribution of $(S_{t})_{t\in[0,T]}$
under $\mathbb{Q}$ is equal to the distribution of
$\left(S_{t}+\sigma\int_{0}^{t}\theta_{s}ds\right)_{t\in[0,T]}$ under
$\mathbb{P}$. Moreover,

$\mathbb{E}_{\mathbb{Q}}\left[\frac{1}{\alpha}\log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)\right]=\mathbb{E}_{\mathbb{Q}}\left[\frac{1}{2\alpha}\int_{0}^{T}\tau^{2}_{t}(Y)dt\right]=\mathbb{E}_{\mathbb{P}}\left[\frac{1}{2\alpha}\int_{0}^{T}\theta^{2}_{t}dt\right].$

Finally, choose $M^{\mathbb{Q}}\in\mathcal{M}^{\mathbb{Q}}$ such that the law
of $(W^{\mathbb{Q}},M^{\mathbb{Q}})$ under $\mathbb{Q}$ is equal to the law of
$(W,M+\sigma W)$ under $\mathbb{P}$. We conclude,

$\displaystyle\mathbb{E}_{\mathbb{Q}}\left[X-\frac{1}{\alpha}\int_{0}^{T}\log\left(\frac{d\mathbb{Q}}{d\mathbb{P}}\right)-\Phi_{0}(M^{\mathbb{Q}}_{0}-S_{0})-\frac{1}{2\Lambda}\int_{0}^{T}|M^{\mathbb{Q}}_{t}-S_{t}|^{2}dt\right]$
$\displaystyle=\mathbb{E}_{\mathbb{P}}\left[f\left(S_{T}+\sigma\int_{0}^{T}\theta_{t}dt\right)-\frac{1}{2\alpha}\int_{0}^{T}\theta^{2}_{t}dt\right.$
$\displaystyle\left.-\Phi_{0}(M_{0}-S_{0})-\frac{1}{2\Lambda}\int_{0}^{T}\left|S_{0}+\mu
t+\sigma\int_{0}^{t}\theta_{s}ds-M_{t}\right|^{2}dt\right].$

This together with (3.1) gives (3) as required. ∎

Next, we prove the following.

######  Lemma 3.3.

Denote by $L^{2}_{0}(\mathcal{F}_{T},\mathbb{P})$ the set of all random
variables of the form

(3.4) $Z=\iota+\int_{0}^{T}\kappa_{t}dW_{t}$

for some $\iota\in\mathbb{R}$ and a predictable and bounded process
$\kappa=(\kappa_{t})_{t\in[0,T]}$ such that $\kappa_{[T-\epsilon,T]}\equiv 0$
for some (deterministic) $\epsilon>0$. Let $Z\in
L^{2}_{0}(\mathcal{F}_{T},\mathbb{P})$. There exists a constant $\hat{C}>0$
(may depend on $Z$) such that for any $\Lambda\in(0,1)$

$\displaystyle\sup_{(\theta,M)\in\Gamma\times\mathcal{M}}\mathbb{E}_{\mathbb{P}}\left[f\left(S_{T}+\sigma\int_{0}^{T}\theta_{t}dt\right)-\frac{1}{2\alpha(\Lambda)}\int_{0}^{T}\theta^{2}_{t}dt\right.$
$\displaystyle\left.-\Phi_{0}(M_{0}-S_{0})-\frac{1}{2\Lambda}\int_{0}^{T}\left|S_{0}+\mu
t+\sigma\int_{0}^{t}\theta_{s}ds-M_{t}\right|^{2}dt\right]$
$\displaystyle\geq\mathbb{E}_{\mathbb{P}}\left[f\left(S_{0}+\sigma
W_{T}+Z\right)-\frac{\left(Z+\sigma\sqrt{A}\Phi_{0}\right)^{2}}{4\sigma\sqrt{A}}\right]+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}-\hat{C}\Lambda$

where, as before $\alpha(\Lambda)=\frac{A}{\Lambda}$.

###### Proof.

Let $Z$ given by (3.4) and let $\Xi$ be the map from Proposition 5.1. Define
the deterministic function $\nu:[0,T]\rightarrow\mathbb{R}$ by
$\nu:=\Xi_{T}(\Lambda,\iota,\Phi_{0})$ and for any $s<T$ define the stochastic
process $(l_{\cdot,s})_{\cdot\in[s,T]}$ by
$(l_{\cdot,s})_{\cdot\in[s,T]}=\Xi_{T-s}(\Lambda,\kappa_{s},0)$.

Next, introduce $(\theta,M)\in\Gamma\times\mathcal{M}$

$\displaystyle\theta_{t}:=\frac{\dot{\nu}_{t}-\mu}{\sigma}+\frac{1}{\sigma}\int_{0}^{t}\frac{\partial
l_{t,s}}{\partial t}dW_{s},\ \ t\in[0,T],$ $\displaystyle
M_{t}:=S_{0}+\frac{\int_{0}^{T}\nu_{t}dt-\Phi_{0}\Lambda}{T}+\sigma\int_{0}^{t}\left(1+\frac{1}{T-s}\int_{s}^{T}l_{t,s}dt\right)dW_{s},\
\ t\in[0,T].$

Observe that from the definition of $\Xi$ we have

$\nu_{0}=0,\ \ \nu_{T}=\iota\ \ \mbox{and}\ \ l_{s,s}=0,\ \
l_{T,s}=\kappa_{s}\ \ \forall s.$

This together with the Fubini theorem, the Itô Isometry, (2.1) and (3.4) gives

$\displaystyle\mathbb{E}_{\mathbb{P}}\left[f\left(S_{T}+\sigma\int_{0}^{T}\theta_{t}dt\right)-\frac{1}{2\alpha(\Lambda)}\int_{0}^{T}\theta^{2}_{t}dt-\Phi_{0}(M_{0}-S_{0})-\right.$
$\displaystyle\left.\frac{1}{2\Lambda}\int_{0}^{T}\left|S_{0}+\mu
t+\sigma\int_{0}^{t}\theta_{s}ds-M_{t}\right|^{2}dt\right]$ (3.5)
$\displaystyle=\mathbb{E}_{\mathbb{P}}\left[f\left(S_{0}+\sigma
W_{T}+Z\right)\right]$
$\displaystyle+\frac{\mu\Lambda\iota}{\sigma^{2}A}-\frac{\mu^{2}\Lambda}{2\sigma^{2}A}-I(\Lambda,\nu)-\int_{0}^{T-\epsilon}\mathbb{E}_{\mathbb{P}}\left[J_{s}(\Lambda,l)\right]ds$

where

$I(\Lambda,\nu):=\frac{\Lambda}{2\sigma^{2}A}\int_{0}^{T}\dot{\nu}_{t}^{2}dt+\frac{1}{2\Lambda}\left(\int_{0}^{T}\nu^{2}_{t}dt-\frac{1}{T}\left(\Phi_{0}\Lambda-\int_{0}^{T}\nu_{t}dt\right)^{2}\right)$

and

$J_{s}(\Lambda,l):=\frac{\Lambda}{2\sigma^{2}A}\int_{s}^{T}\left(\frac{\partial
l_{t,s}}{\partial
t}\right)^{2}dt+\frac{1}{2\Lambda}\left(\int_{s}^{T}l^{2}_{t,s}dt-\frac{1}{T-s}\left(\int_{s}^{T}l_{t,s}dt\right)^{2}\right).$

From Proposition 5.1 there exists a constant $C>0$ (may depend on $\iota$ and
$\kappa$) such that

(3.6)
$\left|I(\Lambda,\nu)-\frac{\left(\iota+\sigma\sqrt{A}\Phi_{0}\right)^{2}}{4\sigma\sqrt{A}}+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}\right|\leq
C\Lambda$

and for any $s\in[0,T-\epsilon]$

(3.7)
$\left|J_{s}(\Lambda,l)-\frac{\kappa^{2}_{s}}{4\sigma\sqrt{A}}\right|\leq
C\Lambda.$

By combining the Itô Isometry and (3)–(3.7) we complete the proof. ∎

We now have all the pieces in place that we need for the completion of the
proof of Proposition 3.1.

###### Proof.

Recall the definition of $g$ given in (2.5). From the Lipschitz continuity of
$f$ it follows that there exists a bounded (measurable) function
$\zeta:\mathbb{R}\rightarrow\mathbb{R}$ such that

(3.8) $g(x)=f\left(x+\zeta(x)\right)-\frac{\zeta^{2}(x)}{4\sigma\sqrt{A}},\ \
\ \forall x\in\mathbb{R}.$

Choose a sequence $Z_{n}\in L^{2}_{0}(\mathcal{F}_{T},\mathbb{P})$,
$n\in\mathbb{N}$ such that

$\lim_{n\rightarrow\infty}Z_{n}=\zeta(S_{0}-\sigma\sqrt{A}\Phi_{0}+\sigma
W_{T})-\sigma\sqrt{A}\Phi_{0}$

where the limit is in $L^{2}(\mathbb{P}).$ From Lemmas 3.2–3.3 and (3.8) we
obtain

$\displaystyle\lim\inf_{\Lambda\downarrow
0}c\left(\Lambda,A/\Lambda,\Phi_{0},X\right)$
$\displaystyle\geq\sup_{n\in\mathbb{N}}\mathbb{E}_{\mathbb{P}}\left[f\left(S_{0}+\sigma
W_{T}+Z_{n}\right)-\frac{\left(Z_{n}+\sigma\sqrt{A}\Phi_{0}\right)^{2}}{4\sigma\sqrt{A}}\right]+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}$
$\displaystyle\geq\mathbb{E}_{\mathbb{P}}\left[g\left(S_{0}-\sigma\sqrt{A}\Phi_{0}+\sigma
W_{T}\right)\right]+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}$
$\displaystyle=u\left(0,S_{0}-\sigma\sqrt{A}\Phi_{0}\right)+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}.$

∎

##  4\. Proof of the Upper Bound

In order to complete the proof of Theorem 2.1 it remains to establish the
following result.

######  Proposition 4.1.

Recall the trading strategies $\Phi^{\Lambda}$, $\Lambda>0$ given by (2.8).
Then,

$\lim\sup_{\Lambda\downarrow
0}\frac{\Lambda}{A}\log\left(\mathbb{E}_{\mathbb{P}}\left[\exp\left(\frac{A}{\Lambda}\left(X-V^{\Phi^{\Lambda}}_{T}\right)\right)\right]\right)\leq
u\left(0,S-\sigma\sqrt{A}\Phi_{0}\right)+\frac{\sigma\sqrt{A}\Phi^{2}_{0}}{2}.$

###### Proof.

The proof will be done in three steps.
Step I:  In this step we use the specific structure of the payoff $f$ given by
(2.3). Let us show that for any $\Lambda>0$

(4.1) $g\left(S_{T}-\sigma\sqrt{A}F^{\Lambda}_{T}\right)\geq
f(S_{T})-\frac{\sigma\sqrt{A}|\Phi_{0}\Theta|}{\sinh\left(\sqrt{\rho(\Lambda)}T\right)}$

where, as before $\rho(\Lambda):=\frac{\sigma^{2}A}{\Lambda^{2}}$.

Fix $\Lambda>0$. From (2.7)

$\frac{d}{dt}\left[\frac{F^{\Lambda}_{t}}{\cosh\left(\sqrt{\rho(\Lambda)}(T-t)\right)}\right]=\frac{\sqrt{\rho(\Lambda)}}{2\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)}\Upsilon^{\Lambda}_{t},\
\ \ t\in[0,T]$

where, recall that $\Upsilon^{\Lambda}_{t}:=\frac{\partial u}{\partial
x}\left(t,S_{t}-\sigma\sqrt{A}F^{\Lambda}_{t}\right)$, $t\in[0,T)$. Clearly,
$|\Upsilon^{\Lambda}_{t}|\leq\Theta$, and so,

$\displaystyle|F^{\Lambda}_{T}|\leq\left|\frac{F^{\Lambda}_{0}}{\cosh\left(\sqrt{\rho(\Lambda)}T\right)}\right|+\Theta\int_{0}^{T}\frac{\sqrt{\rho(\Lambda)}}{2\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)}dt$
$\displaystyle\leq\left|\frac{\Phi_{0}}{\sinh\left(\sqrt{\rho(\Lambda)}T\right)}\right|+|\Theta|.$

This together with (2.3) and (2.5) gives (4.1).
Step II: In this step we prove that there exists a constant $\tilde{C}>0$ such
tht

(4.2)
$\left|\int_{0}^{T}\Upsilon^{\Lambda}_{t}dt-\int_{0}^{T}\Phi^{\Lambda}_{t}dt\right|\leq\tilde{C}\Lambda,\
\ \forall\Lambda>0.$

Fix $\Lambda>0$. From (2.11)

$\displaystyle\frac{d}{dt}\left[\frac{\Phi^{\Lambda}_{t}}{\cosh\left(\sqrt{\rho(\Lambda)}(T-t)\right)}\right]$
$\displaystyle=\frac{\sqrt{\rho(\Lambda)}}{2\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)}\tanh\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)\Upsilon^{\Lambda}_{t},\
\ \ t\in[0,T].$

We get

$\displaystyle\Phi^{\Lambda}_{t}=\Phi_{0}\frac{\cosh\left(\sqrt{\rho(\Lambda)}(T-t)\right)}{\cosh(\sqrt{\rho(\Lambda)}T)}$
$\displaystyle+\int_{0}^{t}\frac{\sqrt{\rho(\Lambda)}\cosh\left(\sqrt{\rho(\Lambda)}(T-t)\right)}{2\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-s)}{2}\right)}\tanh\left(\frac{\sqrt{\rho(\Lambda)}(T-s)}{2}\right)\Upsilon^{\Lambda}_{s}ds$

and so, from the Fubini theorem

$\int_{0}^{T}\Phi^{\Lambda}_{t}dt-\int_{0}^{T}\Upsilon^{\Lambda}_{t}dt=\Phi_{0}\frac{\tanh\left(\sqrt{\rho(\Lambda)}T\right)}{\sqrt{\rho(\Lambda)}}-\int_{0}^{T}\frac{\Upsilon^{\Lambda}_{s}}{\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-s)}{2}\right)}ds.$

This together with the simple integral

$\int_{0}^{T}\frac{ds}{\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-s)}{2}\right)}=\frac{2\tanh\left(\frac{\sqrt{\rho(\Lambda)}T}{2}\right)}{\sqrt{\rho(\Lambda)}}$

and the inequality $|\Upsilon^{\Lambda}_{t}|\leq\Theta$ gives (4.2).
Step III: In this step we complete the proof. Fix $\Lambda>0$ and introduce
the process

$M^{\Lambda}_{t}:=\exp\left(\frac{A}{\Lambda}\left(u\left(t,S_{t}-\sigma\sqrt{A}F^{\Lambda}_{t}\right)+\frac{\sigma\sqrt{A}F^{\Lambda}_{t}\Phi^{\Lambda}_{t}}{2}-V^{\Phi^{\Lambda}}_{t}\right)\right),\
\ t\in[0,T].$

From the Itô formula, (2.2), (2.6)–(2.8) and (2.11) we obtain

$\displaystyle\frac{dM^{\Lambda}_{t}}{M^{\Lambda}_{t}}=\frac{A}{\Lambda}\left(\Upsilon^{\Lambda}_{t}-\Phi^{\Lambda}_{t}\right)dS_{t}+\frac{\sigma^{2}A^{2}}{2\Lambda^{2}}\left(\Upsilon^{\Lambda}_{t}-\Phi^{\Lambda}_{t}\right)^{2}dt$
$\displaystyle-\frac{\sigma^{2}A^{2}}{\Lambda^{2}}\Upsilon^{\Lambda}_{t}\left(\frac{\cosh\left(\sqrt{\rho(\Lambda)}(T-t)\right)}{2\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)}\Upsilon^{\Lambda}_{t}-\Phi^{\Lambda}_{t}\right)dt$
$\displaystyle+\frac{\sigma^{2}A^{2}}{2\Lambda^{2}}\left(\tanh\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)\Upsilon^{\Lambda}_{t}-\coth\left(\sqrt{\rho(\Lambda)}(T-t)\right)\Phi^{\Lambda}_{t}\right)^{2}dt$
$\displaystyle+\frac{\sigma^{2}A^{2}}{2\Lambda^{2}}\Phi^{\Lambda}_{t}\left(\frac{\cosh\left(\sqrt{\rho(\Lambda)}(T-t)\right)}{2\cosh^{2}\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)}\Upsilon^{\Lambda}_{t}-\Phi^{\Lambda}_{t}\right)dt$
$\displaystyle+\frac{\sigma^{2}A^{2}}{2\Lambda^{2}}\coth\left(\sqrt{\rho(\Lambda)}(T-t)\right)\Phi^{\Lambda}_{t}$
$\displaystyle\times\left(\tanh\left(\frac{\sqrt{\rho(\Lambda)}(T-t)}{2}\right)\Upsilon^{\Lambda}_{t}-\coth\left(\sqrt{\rho(\Lambda)}(T-t)\right)\Phi^{\Lambda}_{t}\right)$
$\displaystyle=\frac{A}{\Lambda}\left(\Upsilon^{\Lambda}_{t}-\Phi^{\Lambda}_{t}\right)dS_{t}$

where the last equality follows from simple calculations.

Hence, from (2.1) it follows that the process

$N^{\Lambda}_{t}:=\exp\left(-\frac{\mu
A\int_{0}^{t}\left(\Upsilon^{\Lambda}_{t}-\Phi^{\Lambda}_{s}\right)ds}{\Lambda}\right)M^{\Lambda}_{t},\
\ t\in[0,T]$

is a local–martingale, and so from the obvious inequality $N^{\Lambda}>0$ we
conclude that this process is a super–martingale.

Finally,

$\displaystyle\frac{\Lambda}{A}\log\left(\mathbb{E}_{\mathbb{P}}\left[\exp\left(\frac{A}{\Lambda}\left(X-V^{\Phi^{\Lambda}}_{T}\right)\right)\right]\right)$
$\displaystyle\leq\frac{\Lambda}{A}\log\left(\mathbb{E}_{\mathbb{P}}[M^{\Lambda}_{T}]\right)+\frac{\sigma\sqrt{A}|\Phi_{0}\Theta|}{\sinh\left(\sqrt{\rho(\Lambda)}T\right)}$
$\displaystyle\leq\frac{\Lambda}{A}\log\left(\mathbb{E}_{\mathbb{P}}[N^{\Lambda}_{T}]\right)+\tilde{C}|\mu|\Lambda+\frac{\sigma\sqrt{A}|\Phi_{0}\Theta|}{\sinh\left(\sqrt{\rho(\Lambda)}T\right)}$
$\displaystyle\leq\frac{\Lambda}{A}\log\left(N^{\Lambda}_{0}\right)+\tilde{C}|\mu|\Lambda+\frac{\sigma\sqrt{A}|\Phi_{0}\Theta|}{\sinh\left(\sqrt{\rho(\Lambda)}T\right)}$
$\displaystyle=u\left(0,S_{0}-\sigma\sqrt{A}\Phi_{0}\coth\left(\sqrt{\rho(\Lambda)}T\right)\right)+\frac{\sigma\sqrt{A}\Phi^{2}_{0}\coth\left(\sqrt{\rho(\Lambda)}T\right)}{2}$
$\displaystyle+\tilde{C}|\mu|\Lambda+\frac{\sigma\sqrt{A}|\Phi_{0}\Theta|}{\sinh\left(\sqrt{\rho(\Lambda)}T\right)}.$

The first inequality follows from (4.1) and the relations
$u(T,\cdot)=g(\cdot)$, $\Phi^{\Lambda}_{T}=0$. The second inequality is due to
(4.2). The super–martingale property of $N^{\Lambda}$ gives the third
inequality. The equality is due to (2.8).

By taking $\Lambda\downarrow 0$ we complete the proof. ∎

##  5\. Auxiliary Result

For any $\mathbb{T}\in(0,T]$ and $x\in\mathbb{R}$ let $C_{0,x}[0,\mathbb{T}]$
be the space of all continuous functions
$z:[0,\mathbb{T}]\rightarrow\mathbb{R}$ which satisfy $z_{0}=0$ and
$z_{\mathbb{T}}=x$.

######  Proposition 5.1.

For any $\mathbb{T}\in(0,T]$ there exists a measurable map
$\Xi_{\mathbb{T}}:(0,1)\times\mathbb{R}^{2}\rightarrow C[0,\mathbb{T})$ such
that for any $\Lambda\in(0,1)$ and $x,\phi\in\mathbb{R}$ the continuous
function $\Xi_{\mathbb{T}}(\Lambda,x,\phi)\in C_{0,x}[0,\mathbb{T}]$ is the
unique minimizer for the optimization problem

(5.1) $\min_{\delta\in
C_{0,x}[0,\mathbb{T}]}\left[\frac{\Lambda}{2\sigma^{2}A}\int_{0}^{\mathbb{T}}\dot{\delta}_{t}^{2}dt+\frac{1}{2\Lambda}\left(\int_{0}^{\mathbb{T}}\delta^{2}_{t}dt-\frac{1}{\mathbb{T}}\left(\phi\Lambda-\int_{0}^{\mathbb{T}}\delta_{t}dt\right)^{2}\right)\right].$

Moreover, denote the corresponding value by $V_{\mathbb{T}}(\Lambda,x,\phi)$.
Then, for any $\epsilon>0$ and a compact set $K\subset\mathbb{R}^{2}$ there
exists a constant $\hat{C}$ (may depend on $\epsilon$ and $K$) such that

(5.2)
$\left|V_{\mathbb{T}}(\Lambda,x,\phi)-\frac{\left(x+\sigma\sqrt{A}\phi\right)^{2}}{4\sigma\sqrt{A}}+\frac{\sigma\sqrt{A}\phi^{2}}{2}\right|\leq\hat{C}\Lambda,\
\ \forall(\mathbb{T},\Lambda,x,\phi)\in[\epsilon,T]\times(0,1)\times K.$

###### Proof.

Fix
$(\mathbb{T},\Lambda,x,\phi)\in[\epsilon,T]\times(0,1)\times\mathbb{R}^{2}$.
First we minimize the pattern given by (5.1) under the additional constraint
that $\int_{0}^{\mathbb{T}}\delta_{t}dt$ is given. Then, we will find the
optimal $\int_{0}^{\mathbb{T}}\delta_{t}dt$.

For any $y\in\mathbb{R}$ let $C^{y}_{0,x}[0,\mathbb{T}]\subset
C_{0,x}[0,\mathbb{T}]$ be the subset of all functions $\delta\in
C_{0,x}[0,\mathbb{T}]$ which satisfy $\int_{0}^{\mathbb{T}}\delta_{t}dt=y$.
Consider the minimization problem

$\min_{\delta\in
C^{y}_{0,x}[0,\mathbb{T}]}\int_{0}^{\mathbb{T}}H(\dot{\delta}_{t},\delta_{t})dt$

where
$H(v_{1},v_{2}):=\frac{\Lambda}{2\sigma^{2}A}v^{2}_{1}+\frac{1}{2\Lambda}v^{2}_{2}$
for $v_{1},v_{2}\in\mathbb{R}$. This optimization problem is convex and so it
has a unique solution which has to satisfy the Euler–Lagrange equation (for
details see Gelfand & Fomin (1963)) $\frac{d}{dt}\frac{\partial
H}{\partial\dot{\delta}_{t}}=\lambda+\frac{d}{dt}\frac{\partial
H}{\partial\delta_{t}}$ for some constant $\lambda>0$ (lagrange multiplier due
to the constraint $\int_{0}^{\mathbb{T}}\delta_{t}dt=y$). Thus, the optimizer
which we denote by $\hat{\delta}$ solves the ODE
$\ddot{\hat{\delta}}_{t}-\rho\hat{\delta}\equiv const$ (recall the risk-
liquidity ration $\rho=\rho(\Lambda):=\frac{\sigma^{2}A}{\Lambda^{2}}$). From
the standard theory it follows that

(5.3)
$\hat{\delta}_{t}=c_{1}\sinh(\sqrt{\rho}t)+c_{2}\sinh(\sqrt{\rho}(T-t))+c_{3},\
\ t\in[0,\mathbb{T}]$

for some constants $c_{1},c_{2},c_{3}$. From the three constraints
$\hat{\delta}_{0}=0$, $\hat{\delta}_{\mathbb{T}}=x$ and
$\int_{0}^{\mathbb{T}}\hat{\delta}_{t}dt=y$ we obtain

(5.4) $c_{1}=\frac{x-c_{3}}{\sinh(\sqrt{\rho}\mathbb{T})},\ \
c_{2}=-\frac{c_{3}}{\sinh(\sqrt{\rho}\mathbb{T})}\ \ \mbox{and}\ \
c_{3}=\frac{\sqrt{\rho}y-x\tanh(\sqrt{\rho}\mathbb{T}/2)}{\sqrt{\rho}\mathbb{T}-2\tanh(\sqrt{\rho}\mathbb{T}/2)}.$

We argue that

$\displaystyle\rho\int_{0}^{\mathbb{T}}\hat{\delta}^{2}_{t}dt+\int_{0}^{\mathbb{T}}\dot{\hat{\delta}}^{2}_{t}dt=\rho\int_{0}^{\mathbb{T}}\left((\hat{\delta}_{t}-c_{3})+c_{3}\right)^{2}dt+\int_{0}^{\mathbb{T}}\dot{\hat{\delta}}^{2}_{t}dt$
$\displaystyle=\frac{\sqrt{\rho}}{2}\left(c^{2}_{1}+c^{2}_{2}\right)\sinh\left(2\sqrt{\rho}\mathbb{T}\right)-2c_{1}c_{2}\sqrt{\rho}\sinh(\sqrt{\rho}\mathbb{T})-\rho
c^{2}_{3}\mathbb{T}+2\rho c_{3}y$
$\displaystyle=\sqrt{\rho}x^{2}\coth(\sqrt{\rho}\mathbb{T})+2\sqrt{\rho}c_{1}c_{2}\sinh(\sqrt{\rho}\mathbb{T})\left(\cosh(\sqrt{\rho}\mathbb{T})-1\right)-\rho
c^{2}_{3}\mathbb{T}+2\rho c_{3}y$
$\displaystyle=\sqrt{\rho}x^{2}\coth(\sqrt{\rho}\mathbb{T})+\left(2\sqrt{\rho}\tanh(\sqrt{\rho}\mathbb{T}/2)-\rho\mathbb{T}\right)c^{2}_{3}$
$\displaystyle+2\left(\rho
y-\sqrt{\rho}\tanh(\sqrt{\rho}\mathbb{T}/2)x\right)c_{3}$ (5.5)
$\displaystyle=\sqrt{\rho}\left(x^{2}\coth(\sqrt{\rho}\mathbb{T})+\frac{\left(x\tanh(\sqrt{\rho}\mathbb{T}/2)-\sqrt{\rho}y\right)^{2}}{\sqrt{\rho}\mathbb{T}-2\tanh(\sqrt{\rho}\mathbb{T}/2)}\right).$

Indeed, the first equality is obvious. The second equality follows from (5.3)
and simple computations. The third equality is due to
$c_{1}-c_{2}=\frac{x}{\sinh(\sqrt{\rho}\mathbb{T})}$. The fourth equality is
due to $c_{1}c_{2}=\frac{c^{2}_{3}-xc_{3}}{\sinh^{2}(\sqrt{\rho}\mathbb{T})}$.
The last equality follows from substituting $c_{3}$.

From (5) we conclude that in order to minimize (5.1) we need to find $y$ which
minimizes the quadratic pattern

$\frac{1}{2\sqrt{\rho}\Lambda}\
\frac{\left(x\tanh(\sqrt{\rho}\mathbb{T}/2)-\sqrt{\rho}y\right)^{2}}{\sqrt{\rho}\mathbb{T}-2\tanh(\sqrt{\rho}\mathbb{T}/2)}-\frac{1}{2\Lambda\mathbb{T}}\left(\phi\Lambda-y\right)^{2}.$

Observe that this quadratic pattern is convex in $y$ and so has a unique
minimum

(5.6)
$y=\frac{x\mathbb{T}}{2}-\frac{\phi\Lambda\left(\sqrt{\rho}\mathbb{T}-2\tanh(\sqrt{\rho}\mathbb{T}/2)\right)}{2\tanh(\sqrt{\rho}\mathbb{T}/2)}.$

Thus, define $\Xi_{\mathbb{T}}(\Lambda,x,\phi):=\hat{\delta}$ where
$\hat{\delta}$ is given by (5.3)–(5.4) and (5.6). Clearly,
$\Xi_{\mathbb{T}}(\Lambda,x,\phi)$ is the unique minimizer for (5.1).

Let

$V_{\mathbb{T}}(\Lambda,x,\phi):=\frac{\Lambda}{2\sigma^{2}A}\int_{0}^{\mathbb{T}}\dot{\hat{\delta}}_{t}^{2}dt+\frac{1}{2\Lambda}\left(\int_{0}^{\mathbb{T}}\hat{\delta}^{2}_{t}dt-\frac{1}{\mathbb{T}}\left(\phi\Lambda-\int_{0}^{\mathbb{T}}\hat{\delta}_{t}dt\right)^{2}\right).$

Finally, we prove (5.2). Choose $\epsilon>0$ and a compact set
$K\subset\mathbb{R}^{2}$. Assume that
$(\mathbb{T},x,\phi)\in[\epsilon,T]\times K$. From (5) and the equality
$\rho=\frac{\sigma^{2}A}{\Lambda^{2}}$ we get that there exists a constant
$C_{1}$ (may depend on $\epsilon$ and $K$) such that

(5.7)
$\left|V_{\mathbb{T}}(\Lambda,x,\phi)-\left(\frac{x^{2}}{2\sigma\sqrt{A}}+\frac{y^{2}}{\sigma\sqrt{A}\mathbb{T}^{2}}+\frac{\phi
y}{\mathbb{T}}-\frac{xy}{\sigma\sqrt{A}\mathbb{T}}\right)\right|\leq
C_{1}\Lambda$

where $y$ given by (5.6). From (5.6) we have
$\left|y-\frac{\mathbb{T}}{2}\left(x-\sigma\sqrt{A}\phi\right)\right|\leq
C_{2}\Lambda$ for some constant $C_{2}$ (may depend on $\epsilon$ and $K$).
This together with (5.7) gives (5.2) and completes the proof. ∎

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