# Multi-asset optimal execution and statistical arbitrage strategies under
Ornstein-Uhlenbeck dynamics

Philippe Bergault111Université Paris 1 Panthéon-Sorbonne, Centre d’Economie de
la Sorbonne, 106 Boulevard de l’Hôpital, 75642 Paris Cedex 13, France,
<EMAIL_ADDRESS>Fayçal Drissi222Université Paris 1
Panthéon-Sorbonne, Centre d’Economie de la Sorbonne, 106 Boulevard de
l’Hôpital, 75642 Paris Cedex 13, France<EMAIL_ADDRESS>Olivier Guéant333Université Paris 1 Panthéon-Sorbonne, Centre d’Economie de la
Sorbonne, 106 Boulevard de l’Hôpital, 75642 Paris Cedex 13, France,
<EMAIL_ADDRESS>Member of the Louis Bachelier Fellows.
Corresponding author.

###### Abstract

In recent years, academics, regulators, and market practitioners have
increasingly addressed liquidity issues. Amongst the numerous problems
addressed, the optimal execution of large orders is probably the one that has
attracted the most research works, mainly in the case of single-asset
portfolios. In practice, however, optimal execution problems often involve
large portfolios comprising numerous assets, and models should consequently
account for risks at the portfolio level. In this paper, we address multi-
asset optimal execution in a model where prices have multivariate Ornstein-
Uhlenbeck dynamics and where the agent maximizes the expected (exponential)
utility of her PnL. We use the tools of stochastic optimal control and
simplify the initial multidimensional Hamilton-Jacobi-Bellman equation into a
system of ordinary differential equations (ODEs) involving a Matrix Riccati
ODE for which classical existence theorems do not apply. By using a priori
estimates obtained thanks to optimal control tools, we nevertheless prove an
existence and uniqueness result for the latter ODE, and then deduce a
verification theorem that provides a rigorous solution to the execution
problem. Using examples based on data from the foreign exchange and stock
markets, we eventually illustrate our results and discuss their implications
for both optimal execution and statistical arbitrage.

Key words: Optimal execution, Statistical arbitrage, Stochastic optimal
control, Riccati equations.

##  1 Introduction

When executing large blocks of assets, financial agents need to control their
overall trading costs by finding the optimal balance between trading rapidly
to mitigate market price risk and trading slowly to minimize execution costs
and market impact. Building on the first rigorous approaches introduced by
Bertsimas and Lo in [11] and Almgren and Chriss in [6] and [7], many models
for the optimal execution of large orders have been proposed in the last two
decades. Subsequently, almost all practitioners today slice their large orders
into small (child) orders according to optimized trading schedules inspired by
the academic literature.

The basic Almgren-Chriss model is a discrete-time model where the agent posts
market orders (MOs) to maximize a mean-variance objective function. Many
extensions of this seminal model have been proposed. Regarding the framework,
(Forsyth and Kennedy, [18]) examines the use of quadratic variation rather
than variance in the objective function, (Schied and Schöneborn, [38]) uses
stochastic optimal control tools to characterize and find optimal strategies
for a Von Neumann–Morgenstern investor, and (Guéant, [23]) provides results
for optimal liquidation within a Von Neumann-Morgenstern expected utility
framework with general market impact functions and derives subsequent results
for block trade pricing. As for the model parameters, (Almgren, [3]) studies
the case of random execution costs, (Almgren, [4, 5]) addresses stochastic
liquidity and volatility, (Lehalle, [32]) discusses how to take into account
statistical aspects of the variability of estimators of the main exogenous
variables such as volumes or volatilities in the optimization phase, and
(Cartea and Jaimungal, [14]) provides a closed-form strategy incorporating
order flows from all agents. Furthermore, numerous market impact and limit
order book (LOB) models have also been studied. For instance, (Lorenz and
Schied, [35]) investigates the stability of optimal strategies in the presence
of transient price impact with exponential decay and general dynamics of a
drift in the underlying price process accounting for the price impact of other
agents. (Obizhaeva and Wang, [37]), later generalized in (Alfonsi, Fruth, and
Schied, [1]), proposes a single-asset market impact model where price dynamics
are derived from a dynamic LOB model with resilience, (Alfonsi and Schied,
[2]) derives explicit optimal execution strategies in a discrete-time LOB
model with general shape functions and an exponentially decaying price impact,
(Gatheral, [21]) uses the no-dynamic-arbitrage principle to address the
viability of market impact models, and (Gatheral, Schied, and Slynko, [22])
obtains explicit optimal strategies with a transient market impact in an
expected cost minimization setup. As for order and execution strategy types,
the Almgren-Chriss framework focuses on orders of the Implementation Shortfall
(IS) type with MOs only. Other execution strategies have been studied in the
literature, like Volume-Weighted Average Price (VWAP) orders in (Konishi,
[30]), (Frei and Westray, [19]) and (Guéant and Royer, [27]), but also Target
Close (TC) orders and Percentage of Volume (POV) orders, in (Guéant, [24]).
Besides, several models focusing on optimal execution with limit orders have
been proposed, as in (Bayraktar and Ludkovski, [9]), but also in (Guéant,
Lehalle, and Fernandez-Tapia, [26]) and (Guéant and Lehalle, [25]). Regarding
the existence of several venues, the case of optimal splitting of orders
across different liquidity pools has been addressed in (Laruelle, Lehalle, and
Pages, [31]), in (Cartea, Jaimungal, and Penalva, [15]), and more recently in
(Baldacci and Manziuk, [8]).

Another recent and important stream of the optimal execution literature deals
with adding predictive signals of future price changes.444We consider this
stream of the literature to be closely related to our topic of multi-asset
optimal execution. Indeed, when trading an asset, the dynamics of another
asset within or outside the portfolio can be regarded as a predictive signal
that can enhance the execution process. Typical examples of these signals
include a drift in asset prices, order book imbalances, forecasts of the
future order flow of market participants, and other price-based technical
indicators. The usual formalism in the literature with predictive signals is
to consider Brownian or Black-Scholes dynamics, along with independent mean-
reverting Markov signals. The case of Ornstein-Uhlenbeck-type signals is of
special interest as it usually leads to closed-form formulas. For the
interested reader, we refer to (Belak and Muhle-Karbe, [10]) where the authors
consider optimal execution with general Markov signals and an application to
“target zone models”, and to (Lehalle and Neuman, [34]) and (Neuman and Voß,
[36]) in which the authors provide an optimal trading framework incorporating
Markov signals and a transient market impact.

In practice, operators routinely face the problem of having to execute
simultaneously large orders regarding various assets, such as in block trading
for funds facing large subscriptions or withdrawals, or when considering
multi-asset trades in statistical arbitrage trading strategies. More
generally, banks and market makers manage their (il)liquidity and market risk,
when it comes to executing trades, in the context of a central risk book;
hence the need for multi-asset models. However, in contrast to the single-
asset case, the existing literature on the joint execution scheduling of large
orders in multiple assets, or a single asset inside a multi-asset portfolio,
is rather limited. Besides, most existing papers simply consider correlated
Brownian motions when modelling the joint dynamics of prices. The problem of
using single-asset models or unrealistic multivariate models for portfolio
trading is that the resulting trading curves of individual assets do not
balance well execution costs / market impact with price risk at the portfolio
or strategy level.

The first paper presenting a way to build multi-asset trading curves in an
optimized way is (Almgren and Chriss, [7]). Almgren and Chriss considered
indeed, in an appendix of their seminal paper, a multi-asset extension of
their discrete-time model. A few extensions to this model have been proposed
since then. (Lehalle, [33]) considers adding an inventory constraint to
balance the different portfolio lines during the portfolio execution process.
(Schied and Schöneborn, [39]) shows that when prices follow Bachelier
dynamics, deterministic strategies are optimal for a trader with an
exponential utility objective function. In (Cartea, Jaimungal, and Penalva,
[15]), the authors use stochastic control tools to derive optimal execution
strategies for basic multi-asset trading algorithms such as optimal entry/exit
times and cointegration-based statistical arbitrage. (Bismuth, Guéant, and Pu,
[12]) addresses optimal portfolio liquidation (along with other problems) by
coupling Bayesian learning and stochastic control to derive optimal strategies
under uncertainty on model parameters in the Almgren-Chriss framework.
Regarding the literature around the addition of predictive signals,
(Emschwiller, Petit, and Bouchaud, [17]) extends optimal trading with
Markovian predictors to the multi-asset case, with linear trading costs, using
a mean-field approach that reduces the problem to a single-asset one.

A classical model for the multivariate dynamics of financial variables that
goes beyond that of correlated Brownian motions is the multivariate Ornstein-
Uhlenbeck (multi-OU) model. It is especially attractive because it is
parsimonious, and yet general enough to cover a wide spectrum of multi-
dimensional dynamics. Multi-OU dynamics offer indeed a large coverage since
particular cases include correlated Brownian motions but also cointegrated
dynamics which are heavily used in statistical arbitrage. (Cartea, Gan, and
Jaimungal, [13]) is, to our knowledge, the pioneering paper in the use of the
multi-OU model for the price dynamics in a multi-asset optimal execution
problem. Indeed, the authors proposed an interesting model where the asset
prices have multi-OU dynamics and the agent maximizes an objective function
given by the expectation of the Profit and Loss (PnL) minus a running penalty
related to the instantaneous variance of the portfolio. In their approach, the
problem boils down to a system of ODEs involving a Matrix Riccati ODE for
which the classical existence theorems related to linear-quadratic control
theory apply.

In this paper, we propose a model similar to the one in [13], but where the
objective function is of the Von Neumann-Morgenstern type: an expected
exponential utility of the PnL.555Our model accounts therefore for the risk in
a different manner than the model presented in [15]. Comparisons are difficult
to carry out as risk aversion parameters in the two models have different
meanings. By using classical stochastic optimal control tools we show that the
problem boils down to solving a system of ODEs involving a Matrix Riccati ODE.
However, unlike what happens in [13], the use of an expected exponential
utility framework to account for the risk leads to a Matrix Riccati ODE for
which classical existence theorems do not apply. By using a priori estimates
obtained thanks to optimal control tools, we nevertheless prove an existence
and uniqueness result for the latter ODE and obtain a verification theorem
that provides a rigorous solution to the execution problem.

The main contribution of this paper is therefore to propose a model for multi-
asset portfolio execution under multi-OU price dynamics in an expected utility
framework that accounts for the overall risk associated with the execution
process. We focus on the problem where an agent is in charge of unwinding a
large portfolio, but also illustrate the use of our results for multi-asset
statistical arbitrage purposes.

The remainder of this paper is organized as follows. In Section 2 we present
the optimal execution problem in the form of a stochastic optimal control
problem and show that solving the associated Hamilton-Jacobi-Bellman (HJB)
equation boils down to solving a system of ODEs involving a Matrix Riccati
ODE. We then prove a global existence result for that ODE and eventually
provide a solution to the initial stochastic optimal control problem thanks to
a verification argument. In Section 3, we then illustrate our results on both
real data, from the foreign exchange and stock markets, and simulated data.
Our examples focus on optimal liquidation but we also illustrate and discuss
the use of our results for building statistical arbitrage strategies. The core
of the paper is followed by two appendices: one dedicated to the special case
where the multi-OU dynamics reduce to simple correlated Brownian dynamics and
another dedicated to some form of limit case where execution costs and
terminal penalty are ignored – that limit case being useful to obtain a priori
estimates for our general problem.

##  2 The optimal liquidation problem

###  2.1 Modelling framework and notations

In this paper, we consider a filtered probability space
$\left(\Omega,\mathcal{F},\mathbb{P};\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,T]}\right)$
satisfying the usual conditions. We assume this probability space to be large
enough to support all the processes we introduce.

We consider a market with $d\in\mathbb{N}^{*}$ assets,666We denote by
$\mathbb{N}^{*}$ the set $\mathbb{N}^{*}:=\mathbb{N}\backslash\\{0\\}$ of
positive integers. and a trader wishing to liquidate her portfolio over a
period of time $[0,T]$, with $T>0$. Her inventory process777The superscript ⊺
designates the transpose operator. It transforms here a line vector into a
column vector.
$(q_{t})_{t\in[0,T]}=\left(q^{1}_{t},\ldots,q^{d}_{t}\right)^{\intercal}_{t\in[0,T]}$
evolves as

$\displaystyle dq_{t}=v_{t}dt,$ (1)

with $q_{0}\in\mathbb{R}^{d}$ given, where
$(v_{t})_{t\in[0,T]}=(v^{1}_{t},\ldots,v^{d}_{t})^{\intercal}_{t\in[0,T]}$
represents the trading rate of the trader for each asset.

The fundamental prices of the $d$ assets are modelled as a $d$-dimensional
Ornstein-Uhlenbeck process
$(S_{t})_{t\in[0,T]}=\left(S^{1}_{t},\ldots,S^{d}_{t}\right)^{\intercal}_{t\in[0,T]}$:

$\displaystyle dS_{t}$ $\displaystyle=R(\overline{S}-S_{t})dt+VdW_{t},$ (2)

and we introduce the market price process
$(\tilde{S}_{t})_{t\in[0,T]}=\left(\tilde{S}^{1}_{t},\ldots,\tilde{S}^{d}_{t}\right)^{\intercal}_{t\in[0,T]}$
with dynamics:

$\displaystyle d\tilde{S}_{t}$ $\displaystyle=dS_{t}+Kv_{t}dt,$ (3)

with $S_{0}=\tilde{S}_{0}\in{\mathbb{R}}^{d}$ given, where
$\overline{S}\in{\mathbb{R}}^{d}$, $R\in\mathcal{M}_{d}(\mathbb{R})$,
$V\in\mathcal{M}_{d,k}(\mathbb{R})$, $K\in\mathcal{S}_{d}(\mathbb{R})$,888We
denote by $\mathcal{M}_{d,k}(\mathbb{R})$ the set of $d\times k$ real matrices
and by $\mathcal{M}_{d}(\mathbb{R}):=\mathcal{M}_{d,d}(\mathbb{R})$ the set of
$d\times d$ real square matrices. The set of real symmetric $d\times d$
matrices is denoted by $\mathcal{S}_{d}(\mathbb{R})$. and
$(W_{t})_{t\in[0,T]}=\left(W^{1}_{t},\ldots,W^{k}_{t}\right)^{\intercal}_{t\in[0,T]}$
is a $k$-dimensional standard Brownian motion (with independent coordinates)
for some $k\in\mathbb{N}^{*}$. In these dynamics, the matrix $R$ steers the
deterministic part of the process, $\overline{S}$ represents the unconditional
long-term expectation of $(S_{t})_{t\in[0,T]}$, and $V$ drives the dispersion
(for what follows, we introduce $\Sigma=VV^{\intercal}$ the covariation matrix
of the process). The matrix $K$ represents the linear permanent impact the
agent has on the prices.999It is assumed symmetric to avoid price
manipulation. More precisely, since

$d\tilde{S}_{t}=dS_{t}+Kv_{t}dt=R(\overline{S}-S_{t})dt+Kv_{t}dt+VdW_{t}=R(\overline{S}+K(q_{t}-q_{0})-\tilde{S}_{t})dt+Kv_{t}dt+VdW_{t},$

trading impacts both current market prices and long-term expectations.

Ornstein-Uhlenbeck processes are well suited when prices exhibit mean
reversion and/or when there exist one or several linear combinations of asset
prices that are stationary. In the latter case, we say that the assets
involved in the linear combinations are cointegrated (a situation often
encountered in statistical arbitrage). For more details on cointegration in
continuous time, we refer to (Comte [16]).

Finally, the process $(\tilde{X}_{t})_{t\in[0,T]}$ modelling the trader’s cash
account has the dynamics

$\displaystyle d\tilde{X}_{t}=-v_{t}^{\intercal}\tilde{S}_{t}dt-L(v_{t})dt,$
(4)

with $\tilde{X}_{0}\in\mathbb{R}$ given, where
$L:\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}$ is a function representing the
temporary market impact of trades and/or the execution costs incurred by the
trader (see (Guéant, [24]) for an introduction to this type of models). In
this paper, we only consider the case where $L$ is a positive-definite
quadratic form, i.e.101010The subset of positive-definite and positive semi-
definite matrices of $\mathcal{S}_{d}(\mathbb{R})$ are respectively denoted by
$\mathcal{S}^{++}_{d}(\mathbb{R})$ and $\mathcal{S}^{+}_{d}(\mathbb{R})$.

$L(v)=v^{\intercal}\eta v\quad\text{with
}\quad\eta\in\mathcal{S}^{++}_{d}(\mathbb{R}).$

The trader aims at maximizing the expected utility of her wealth at the end of
the trading window $[0,T]$. This wealth is the sum of the amount
$\tilde{X}_{T}$ on the cash account at time $T$ and the value of the remaining
inventory evaluated here at
$q_{T}^{\intercal}\tilde{S}_{T}-\tilde{\ell}(q_{T})$, where the discount term
$\tilde{\ell}(q_{T})$ applied to the Mark-to-Market (MtM) value of the
remaining assets ($q_{T}^{\intercal}\tilde{S}_{T}$) penalizes any non-zero
terminal position.111111This penalization term relaxes the hard constraint
that imposes $q_{T}=0$ in some liquidation problems. This relaxation enables
to use classical tools of optimal control while the problem with hard
constraint (not addressed in this paper) features a singular boundary
condition that makes the problem more difficult to address mathematically. In
this paper, we only consider the case where $\tilde{\ell}$ is a positive-
definite quadratic form, i.e. $\tilde{\ell}(q)=q^{\intercal}\tilde{\Gamma}q$
with $\tilde{\Gamma}\in\mathcal{S}_{d}^{++}(\mathbb{R})$ (see below for a
stronger assumption on $\tilde{\Gamma}$).

To define the set of admissible controls $\mathcal{A}$, we first introduce a
notion of “linear growth” relevant in our context.

######  Definition 1.

Let $t\in[0,T].$ An $\mathbb{R}^{d}$-valued, $\mathbb{F}$-adapted process
$(\zeta_{s})_{s\in[t,T]}$ is said to satisfy a linear growth condition on
$[t,T]$ with respect to $(S_{s})_{s\in[t,T]}$ if there exists a constant
$C_{t,T}>0$ such that for all $s\in[t,T]$,

$\|\zeta_{s}\|\leq
C_{t,T}\left(1+\underset{\tau\in[t,s]}{\sup}\|S_{\tau}\|\right)\vspace{-0.2cm}$

almost surely.121212In all this paper, $\|.\|$ denotes a fixed norm on
$\mathbb{R}^{d}$ (for instance, the Euclidean norm).

We then define for all $t\in[0,T]$:

$\displaystyle\mathcal{A}_{t}=\left\\{(v_{s})_{s\in[t,T]},\
\mathbb{R}^{d}\textrm{-valued},\ \mathbb{F}\textrm{-adapted, satisfying a
linear growth condition with respect to }(S_{s})_{s\in[t,T]}\right\\},$ (5)

and take $\mathcal{A}:=\mathcal{A}_{0}.$131313We restrict our analysis to
linear growth strategies for mathematical convenience, but we expect the
candidate control to be optimal among a larger class of processes.

Mathematically, the trader therefore wants to solve the dynamic optimization
problem

$\displaystyle\underset{v\in\mathcal{A}}{\sup}\
\mathbb{E}\left[-e^{-\gamma\left(\tilde{X}_{T}+q_{T}^{\intercal}\tilde{S}_{T}-\tilde{\ell}(q_{T})\right)}\right],$
(6)

where $\gamma>0$ is the absolute risk aversion parameter of the trader.

Notice that

$\displaystyle\tilde{X}_{T}+q_{T}^{\intercal}\tilde{S}_{T}-\tilde{\ell}(q_{T})$
$\displaystyle=\tilde{X}_{0}+q_{0}^{\intercal}\tilde{S}_{0}+\int_{0}^{T}q_{t}^{\intercal}d\tilde{S}_{t}-\int_{0}^{T}L(v_{t})dt-\tilde{\ell}(q_{T})$
$\displaystyle=X_{0}+q_{0}^{\intercal}S_{0}+\int_{0}^{T}q_{t}^{\intercal}dS_{t}+\int_{0}^{T}q_{t}^{\intercal}Kv_{t}dt-\int_{0}^{T}L(v_{t})dt-\tilde{\ell}(q_{T})$
$\displaystyle=X_{T}+q_{T}^{\intercal}S_{T}-\tilde{\ell}(q_{T})+\frac{1}{2}q_{T}^{\intercal}Kq_{T}-\frac{1}{2}q_{0}^{\intercal}Kq_{0},$

where $X_{0}=\tilde{X}_{0}$ and the process $(X_{t})_{t\in[0,T]}$ has dynamics

$\displaystyle dX_{t}=-v_{t}^{\intercal}S_{t}dt-L(v_{t})dt.$ (7)

Let us now define the penalty function
$\ell:\mathbb{R}^{d}\rightarrow\mathbb{R}$ by:

$\ell(q)=\tilde{\ell}(q)-\frac{1}{2}q^{\intercal}Kq=q^{\intercal}\tilde{\Gamma}q-\frac{1}{2}q^{\intercal}Kq\quad\forall
q\in\mathbb{R}^{d}.$

In what follows, we assume that $\ell$ is a positive semi-definite quadratic
form, i.e.
$\Gamma=\tilde{\Gamma}-\frac{1}{2}K\in\mathcal{S}_{d}^{+}(\mathbb{R})$.

######  Remark 1.

The assumption on $\tilde{\Gamma}$ is not restrictive as in practice,
$\tilde{\Gamma}$ (and therefore $\Gamma$) is chosen arbitrarily large to
enforce liquidation.

It is then straightforward to see that Problem (6) is equivalent to the
following problem:

$\displaystyle\underset{v\in\mathcal{A}}{\sup}\
\mathbb{E}\left[-e^{-\gamma\left(X_{T}+q_{T}^{\intercal}S_{T}-\ell(q_{T})\right)}\right].$
(8)

It is natural to use the tools of stochastic optimal control to solve the
above dynamic optimization problem. Let us define the value function of the
problem
$u:[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
as

$\displaystyle
u(t,x,q,S)=\underset{v\in\mathcal{A}_{t}}{\sup}\mathbb{E}\left[-e^{-\gamma\left(X^{t,x,S,v}_{T}+(q^{t,q,v}_{T})^{\intercal}S^{t,S}_{T}-\ell(q^{t,q,v}_{T})\right)}\right],$
(9)

where, for
$(t,x,q,S)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
and $v\in\mathcal{A}_{t}$, the processes $(q^{t,q,v}_{s})_{s\in[t,T]}$,
$(S^{t,S}_{s})_{s\in[t,T]}$, and $(X^{t,x,S,v}_{s})_{s\in[t,T]}$ have
respective dynamics

$\begin{array}[]{lcl}dq^{t,q,v}_{s}&=&v_{s}ds,\\\
dS^{t,S}_{s}&=&R(\overline{S}-S^{t,S}_{s})ds+VdW_{s},\\\
dX^{t,x,S,v}_{s}&=&-v_{s}^{\intercal}S^{t,S}_{s}ds-L(v_{s})ds,\end{array}$

with $S^{t,S}_{t}=S$, $q^{t,q,v}_{t}=q$, and $X^{t,x,S,v}_{t}=x$.

###  2.2 Hamilton-Jacobi-Bellman equation

The HJB equation associated with Problem (8) is given by141414$u$ will be
solution to that equation, but as we do not know it yet, we write the equation
with an unknown function $w$.

$\displaystyle 0=\ $
$\displaystyle\partial_{t}w(t,x,q,S)+\underset{v\in\mathbb{R}^{d}}{\sup}\left(-(v^{\intercal}S+L(v))\partial_{x}w(t,x,q,S)+v^{\intercal}\nabla_{q}w(t,x,q,S)\right)$
$\displaystyle+(\overline{S}-S)^{\intercal}R^{\intercal}\nabla_{S}w(t,x,q,S)+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}w(t,x,q,S)\right),$ (10)

for all
$(t,x,q,S)\in[0,T)\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
with the terminal condition

$\displaystyle
w(T,x,q,S)=-e^{-\gamma\left(x+q^{\intercal}S-\ell(q)\right)}\quad\forall(x,q,S)\in\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}.$
(11)

In order to study (10), we are going to use the following ansatz:

$\displaystyle
w(t,x,q,S)=-e^{-\gamma\left(x+q^{\intercal}S+\theta(t,q,S)\right)}\quad\forall(t,x,q,S)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}.$
(12)

The interest of this ansatz is based on the following proposition:

######  Proposition 1.

Let $\tau<T$. If there exists $\theta\in
C^{1,1,2}([\tau,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d},\mathbb{R})$
solution to

$\displaystyle 0=\ $
$\displaystyle\partial_{t}\theta(t,q,S)+\underset{v\in\mathbb{R}^{d}}{\sup}\left(v^{\intercal}\nabla_{q}\theta(t,q,S)-L(v)\right)+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}\theta(t,q,S)\right)$ (13)
$\displaystyle-\frac{\gamma}{2}(q+\nabla_{S}\theta(t,q,S))^{\intercal}\Sigma(q+\nabla_{S}\theta(t,q,S))+(\overline{S}-S)^{\intercal}R^{\intercal}(\nabla_{S}\theta(t,q,S)+q)$

on $[\tau,T)\times\mathbb{R}^{d}\times\mathbb{R}^{d}$, with terminal condition

$\displaystyle\theta(T,q,S)=-\ell(q)\quad\forall(q,S)\in\mathbb{R}^{d}\times\mathbb{R}^{d},$
(14)

then the function
$w:[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
defined by

$w(t,x,q,S)=-e^{-\gamma\left(x+q^{\intercal}S+\theta(t,q,S)\right)}\quad\forall(t,x,q,S)\in[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$

is a solution to (10) on
$[\tau,T)\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ with
terminal condition (11).

###### Proof.

Let $\theta\in
C^{1,1,2}([\tau,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d},\mathbb{R})$ be a
solution to (13) on $[\tau,T)\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ with
terminal condition (14), then we have for all
$(t,x,q,S)\in[\tau,T)\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$:

$\displaystyle\partial_{t}w(t,x,q,S)+\underset{v\in\mathbb{R}^{d}}{\sup}\left(-(v^{\intercal}S+L(v))\partial_{x}w(t,x,q,S)+v^{\intercal}\nabla_{q}w(t,x,q,S)\right)$
$\displaystyle+(\overline{S}-S)^{\intercal}R^{\intercal}\nabla_{S}w(t,x,q,S)+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}w(t,x,q,S)\right)$ $\displaystyle=\ $
$\displaystyle-\gamma\partial_{t}\theta(t,q,S)w(t,x,q,S)+\underset{v\in\mathbb{R}^{d}}{\sup}\left(\gamma(v^{\intercal}S+L(v))w(t,x,q,S)-\gamma
v^{\intercal}(\nabla_{q}\theta(t,q,S)+S)w(t,x,q,S)\right)$
$\displaystyle+\frac{\gamma^{2}}{2}\textrm{Tr}\left(\Sigma(q+\nabla_{S}\theta(t,q,S))(q+\nabla_{S}\theta(t,q,S))^{\intercal}w(t,x,q,S)\right)$
$\displaystyle-\gamma(\overline{S}-S)^{\intercal}R^{\intercal}(\nabla_{S}\theta(t,q,S)+q)w(t,x,q,S)-\frac{1}{2}\textrm{Tr}\left(\gamma\Sigma
D^{2}_{SS}\theta(t,q,S)w(t,x,q,S)\right)$ $\displaystyle=\ $
$\displaystyle-\gamma
w(t,x,q,S)\Bigg{(}\partial_{t}\theta(t,q,S)+\underset{v\in\mathbb{R}^{d}}{\sup}\left(v^{\intercal}\nabla_{q}\theta(t,q,S)-L(v)\right)+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}\theta(t,q,S)\right)$
$\displaystyle-\frac{\gamma}{2}(q+\nabla_{S}\theta(t,q,S))^{\intercal}\Sigma(q+\nabla_{S}\theta(t,q,S))+(\overline{S}-S)^{\intercal}R^{\intercal}(\nabla_{S}\theta(t,q,S)+q)\Bigg{)}$
$\displaystyle=\ $ $\displaystyle 0.$

As it is straightforward to verify that $w$ satisfies the terminal condition
(11), the result is proved. ∎

The above result does not rely on the quadratic assumptions for $L$ and
$\ell$. In the quadratic case we consider in this paper, $\theta$ can be found
in almost closed form. To prove this point, the first thing to notice is that
the Legendre-Fenchel transform of $L$ writes

$\displaystyle H:p\in\mathbb{R}^{d}\mapsto\underset{v\in\mathbb{R}^{d}}{\sup}\
v^{\intercal}p-L(v)=\underset{v\in\mathbb{R}^{d}}{\sup}\
v^{\intercal}p-v^{\intercal}\eta v=\frac{1}{4}p^{\intercal}\eta^{-1}p,$ (15)

as the supremum is reached at $v^{*}=\frac{1}{2}\eta^{-1}p$.

Consequently, we get the following HJB equation for $\theta$:

$\displaystyle 0$
$\displaystyle=\partial_{t}\theta(t,q,S)+\frac{1}{4}\nabla_{q}\theta(t,q,S)^{\intercal}\eta^{-1}\nabla_{q}\theta(t,q,S)+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}\theta(t,q,S)\right)$ (16)
$\displaystyle-\frac{\gamma}{2}(q+\nabla_{S}\theta(t,q,S))^{\intercal}\Sigma(q+\nabla_{S}\theta(t,q,S))+(\overline{S}-S)^{\intercal}R^{\intercal}(\nabla_{S}\theta(t,q,S)+q),$

with terminal condition

$\displaystyle\theta(T,q,S)=-q^{\intercal}\Gamma
q\quad\forall(q,S)\in\mathbb{R}^{d}\times\mathbb{R}^{d}.$ (17)

To further study (16), we introduce a second ansatz and look for a solution
$\theta$ of the following form:

$\displaystyle\theta(t,q,S)=q^{\intercal}A(t)q+q^{\intercal}B(t)S+S^{\intercal}C(t)S+D(t)^{\intercal}q+E(t)^{\intercal}S+F(t)\quad\forall(t,q,S)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
(18)

or equivalently

$\theta(t,q,S)=\begin{pmatrix}q\\\
S\end{pmatrix}^{\intercal}P(t)\begin{pmatrix}q\\\
S\end{pmatrix}+\begin{pmatrix}D(t)\\\
E(t)\end{pmatrix}^{\intercal}\begin{pmatrix}q\\\
S\end{pmatrix}+F(t)\quad\forall(t,q,S)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d},$

where $P:[0,T]\rightarrow\mathcal{S}_{2d}(\mathbb{R})$ is defined as

$\displaystyle P(t)=\begin{pmatrix}A(t)&\frac{1}{2}B(t)\\\
\frac{1}{2}B(t)^{\intercal}&C(t)\end{pmatrix}.$ (19)

The interest of this ansatz is stated in the following proposition:

######  Proposition 2.

Let $\tau<T$. Assume there exist $A\in
C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $B\in
C^{1}\left([\tau,T],\mathcal{M}_{d}(\mathbb{R})\right)$, $C\in
C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $D\in
C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $E\in
C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $F\in
C^{1}\left([\tau,T],\mathbb{R}\right)$ satisfying the system of ODEs

$\displaystyle\begin{cases}A^{\prime}(t)=\frac{\gamma}{2}(B(t)+I_{d})\Sigma(B(t)^{\intercal}+I_{d})-A(t)\eta^{-1}A(t)\\\
B^{\prime}(t)=(B(t)+I_{d})R+2\gamma(B(t)+I_{d})\Sigma
C(t)-A(t)\eta^{-1}B(t)\\\ C^{\prime}(t)=R^{\intercal}C(t)+C(t)R+2\gamma
C(t)\Sigma C(t)-\frac{1}{4}B(t)^{\intercal}\eta^{-1}B(t)\\\
D^{\prime}(t)=-(B(t)+I_{d})R\overline{S}+\gamma(B(t)+I_{d})\Sigma
E(t)-A(t)\eta^{-1}D(t)\\\
E^{\prime}(t)=-2C(t)R\overline{S}+R^{\intercal}E(t)+2\gamma C(t)\Sigma
E(t)-\frac{1}{2}B(t)^{\intercal}\eta^{-1}D(t)\\\
F^{\prime}(t)=-\overline{S}^{\intercal}R^{\intercal}E(t)-Tr(\Sigma
C(t))+\frac{\gamma}{2}E(t)^{\intercal}\Sigma
E(t)-\frac{1}{4}D(t)^{\intercal}\eta^{-1}D(t),\end{cases}$ (20)

where $I_{d}$ denotes the identity matrix in $\mathcal{M}_{d}(\mathbb{R})$,
with terminal conditions

$\displaystyle A(T)=-\Gamma,\quad B(T)=C(T)=D(T)=E(T)=F(T)=0.$ (21)

Then the function $\theta$ defined by (18) satisfies (16) on
$[\tau,T)\times\mathbb{R}^{d}\times\mathbb{R}^{d}$ with terminal condition
(17).

###### Proof.

Let us consider $A\in C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$,
$B\in C^{1}\left([\tau,T],\mathcal{M}_{d}(\mathbb{R})\right)$, $C\in
C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $D\in
C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $E\in
C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $F\in
C^{1}\left([\tau,T],\mathbb{R}\right)$ verifying (20) on $[\tau,T)$ with
terminal condition (21). Let us consider
$\theta:[\tau,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
defined by (18). Then we obtain for all
$(t,q,S)\in[\tau,T)\times\mathbb{R}^{d}\times\mathbb{R}^{d}$:

$\displaystyle\partial_{t}\theta(t,q,S)+\frac{1}{4}\nabla_{q}\theta(t,q,S)^{\intercal}\eta^{-1}\nabla_{q}\theta(t,q,S)+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}\theta(t,q,S)\right)$
$\displaystyle-\frac{\gamma}{2}(q+\nabla_{S}\theta(t,q,S))^{\intercal}\Sigma(q+\nabla_{S}\theta(t,q,S))+(\overline{S}-S)^{\intercal}R^{\intercal}(\nabla_{S}\theta(t,q,S)+q),$
$\displaystyle=\quad$ $\displaystyle
q^{\intercal}A^{\prime}(t)q+q^{\intercal}B^{\prime}(t)S+S^{\intercal}C^{\prime}(t)S+D^{\prime}(t)^{\intercal}q+E^{\prime}(t)^{\intercal}S+F^{\prime}(t)$
$\displaystyle+q^{\intercal}A(t)\eta^{-1}A(t)q+q^{\intercal}A(t)\eta^{-1}B(t)S+\frac{1}{4}S^{\intercal}B(t)^{\intercal}\eta^{-1}B(t)S$
$\displaystyle+D(t)^{\intercal}\eta^{-1}A(t)q+\frac{1}{2}\left(D(t)\right)^{\intercal}\eta^{-1}B(t)S+\frac{1}{4}D(t)^{\intercal}\eta^{-1}D(t)$
$\displaystyle+Tr(\Sigma
C(t))-\frac{\gamma}{2}\left(q+B(t)^{\intercal}q+2C(t)S+E(t)\right)^{\intercal}\Sigma\left(q+B(t)^{\intercal}q+2C(t)S+E(t)\right)$
$\displaystyle+\overline{S}^{\intercal}R^{\intercal}q+\overline{S}^{\intercal}R^{\intercal}\left(B(t)^{\intercal}q+2C(t)S+E(t)\right)-S^{\intercal}R^{\intercal}q-S^{\intercal}R^{\intercal}\left(B(t)^{\intercal}q+2C(t)S+E(t)\right)$
$\displaystyle=\quad$ $\displaystyle
q^{\intercal}\left(A^{\prime}(t)-\frac{\gamma}{2}(B(t)+I_{d})\Sigma(B(t)^{\intercal}+I_{d})+\frac{1}{4}\left(2A(t)\right)\eta^{-1}\left(2A(t)\right)\right)q$
$\displaystyle+q^{\intercal}\left(B^{\prime}(t)-(I_{d}+B(t))R-2\gamma(B(t)+I_{d})\Sigma
C(t)+A(t)\eta^{-1}B(t)\right)S$
$\displaystyle+S^{\intercal}\left(C^{\prime}(t)-R^{\intercal}C(t)-C(t)R-2\gamma
C(t)\Sigma C(t)+\frac{1}{4}B(t)^{\intercal}\eta^{-1}B(t)\right)S$
$\displaystyle+\left(D^{\prime}(t)+(B(t)+I_{d})R\overline{S}-\gamma(B(t)+I_{d})\Sigma
E(t)+A(t)\eta^{-1}D(t)\right)^{\intercal}q$
$\displaystyle+\left(E^{\prime}(t)+2C(t)R\overline{S}-R^{\intercal}E(t)-2\gamma
C(t)\Sigma E(t)+\frac{1}{2}B(t)^{\intercal}\eta^{-1}D(t)\right)^{\intercal}S$
$\displaystyle+\left(F^{\prime}(t)+\overline{S}^{\intercal}R^{\intercal}E(t)+Tr(\Sigma
C(t))-\frac{\gamma}{2}E(t)^{\intercal}\Sigma
E(t)+\frac{1}{4}D(t)^{\intercal}\eta^{-1}D(t)\right)$ $\displaystyle=\quad$
$\displaystyle 0.$

As it is straightforward to verify that $\theta$ satisfies the terminal
condition (17), the result is proved. ∎

######  Remark 2.

Two remarks can be made on the system of ODEs (20):

  * •

This system of ODEs can clearly be decomposed into three groups of equations:
the first three ODEs for $A,$ $B$ and $C$ are independent of the others and
can be solved as a first step; once we know $A,B,$ and $C$ we can solve the
linear ODEs for $D$ and $E$, and finally $F$ can be obtained with a simple
integration;

  * •

When $R=0$ (i.e. in the case where the prices $S$ of the $d$ assets are
correlated arithmetic Brownian motions), there is a trivial solution to the
last five equations which is $B=C=D=E=F=0.$ The function $A$ can then be found
using classical techniques (as shown in Appendix A).

It is noteworthy that the first system, i.e.

$\displaystyle\begin{cases}A^{\prime}(t)=\frac{\gamma}{2}(B(t)+I_{d})\Sigma(B(t)^{\intercal}+I_{d})-A(t)\eta^{-1}A(t)\\\
B^{\prime}(t)=(B(t)+I_{d})R+2\gamma(B(t)+I_{d})\Sigma
C(t)-A(t)\eta^{-1}B(t)\\\ C^{\prime}(t)=R^{\intercal}C(t)+C(t)R+2\gamma
C(t)\Sigma C(t)-\frac{1}{4}B(t)^{\intercal}\eta^{-1}B(t)\end{cases}$ (22)

boils down to the following Matrix Riccati ODE in
$P=\begin{pmatrix}A&\frac{1}{2}B\\\ \frac{1}{2}B&C\end{pmatrix}$:

$\displaystyle P^{\prime}(t)=Q+Y^{\intercal}P(t)+P(t)Y+P(t)UP(t),$ (23)

where

$\displaystyle Q=\ $ $\displaystyle\frac{1}{2}\begin{pmatrix}\gamma\Sigma&R\\\
R^{\intercal}&0\end{pmatrix}\in\mathcal{S}_{2d}(\mathbb{R}),$ $\displaystyle
Y=\ $ $\displaystyle\begin{pmatrix}0&0\\\
\gamma\Sigma&R\end{pmatrix}\in\mathcal{M}_{2d}(\mathbb{R}),$ $\displaystyle
U=\ $ $\displaystyle\begin{pmatrix}-\eta^{-1}&0\\\
0&2\gamma\Sigma\end{pmatrix}\in\mathcal{S}_{2d}(\mathbb{R}),$

and the terminal condition writes

$\displaystyle P(T)=\begin{pmatrix}-\Gamma&0\\\
0&0\end{pmatrix}\in\mathcal{S}_{2d}(\mathbb{R}).$ (24)

When compared to the Matrix Riccati ODEs arising in the linear-quadratic
optimal control literature, the distinctive aspect of our equation is that the
matrix $U$ characterizing the quadratic term in the Riccati equation has both
positive and negative eigenvalues. In particular, we cannot rely on existing
results (see for instance Theorem 3.5 of [20]) to prove that there exists a
solution to (23) with terminal condition (24). In this paper, we address the
existence of a solution by using a priori estimates for the value
function.151515Surprisingly maybe, the inequalities we derive do not seem to
derive in a direct manner from a purely analytic argument such as the
classical comparison principle for matrix Riccati equations – see Theorem 3.4
of [20].

Regarding the set of equations (22), there exists a unique local solution by
Cauchy-Lipschitz theorem. In the following section, we therefore first state a
verification theorem that solves the problem on an interval $[\tau,T]$, and
use that very result to address global existence and uniqueness of a solution
on $[0,T]$.

###  2.3 Main mathematical results

######  Theorem 1.

Let $\tau<T$. Let $A\in
C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $B\in
C^{1}\left([\tau,T],\mathcal{M}_{d}(\mathbb{R})\right)$, $C\in
C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $D\in
C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $E\in
C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $F\in
C^{1}\left([\tau,T],\mathbb{R}\right)$ be a solution to the system (20) on
$[\tau,T)$ with terminal condition (21), and consider the function $\theta$
defined by (18) and the associated function $w$ defined by (12).

Then for all
$(t,x,q,S)\in[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
and $v=(v_{s})_{s\in[t,T]}\in\mathcal{A}_{t}$, we have

$\displaystyle\mathbb{E}\left[-e^{-\gamma\left(X^{t,x,S,v}_{T}+\left(q^{t,q,v}_{T}\right)^{\intercal}S^{t,S}_{T}-\ell(q^{t,q,v}_{T})\right)}\right]\leq
w(t,x,q,S).$ (25)

Moreover, equality is obtained in (25) by taking the optimal control
$(v^{*}_{s})_{s\in[t,T]}\in\mathcal{A}_{t}$ given by the closed-loop feedback
formula

$\displaystyle
v^{*}_{s}=\frac{1}{2}\eta^{-1}\left(2A(s)q^{t,q,v}_{s}+B(s)S^{t,S}_{s}+D(s)\right).$
(26)

In particular, $w=u$ on
$[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}.$

###### Proof.

Let $t\in[\tau,T)$, we first prove that
$(v^{*}_{s})_{s\in[t,T]}\in\mathcal{A}_{t}$ (i.e., $(v^{*}_{s})_{s\in[t,T]}$
is well-defined and admissible). Let us consider the Cauchy initial value
problem

$\displaystyle\forall
s\in[t,T],\quad\frac{d\tilde{q}_{s}}{ds}=\frac{1}{2}\eta^{-1}\left(2A(s)\tilde{q}_{s}+B(s)S^{t,S}_{s}+D(s)\right),\qquad\tilde{q}_{t}=q.$

The unique solution of that Cauchy problem writes

$\displaystyle\tilde{q}_{s}=\exp\left(\int_{t}^{s}\phi(\varrho)d\varrho{}\right)\left(q+\int_{t}^{s}\psi\left(\varrho,S^{t,S}_{\varrho}\right)\exp\left(-\int_{t}^{\varrho}\phi(\varsigma)d\varsigma{}\right)d\varrho\right),$

where $\phi$ and $\psi$ are defined by

$\displaystyle\phi$ $\displaystyle:s\in[t,T]\mapsto\eta^{-1}A(s),$
$\displaystyle\psi$
$\displaystyle:(s,S)\in[t,T]\times\mathbb{R}^{d}\mapsto\frac{1}{2}\eta^{-1}\left(B(s)S+D(s)\right).$

Then $v^{*}$ can be written as

$\displaystyle
v^{*}_{s}=\frac{d\tilde{q}_{s}}{ds}=\phi(s)\exp\left(\int_{t}^{s}\phi(\varrho)d\varrho\right)\left(q+\int_{t}^{s}\psi\left(\varrho,S^{t,S}_{\varrho}\right)\exp\left(-\int_{t}^{\varrho}\phi(\varsigma)d\varsigma{}\right)d\varrho\right)+\psi\left(s,S^{t,S}_{s}\right).$

We see from the definition of $\phi$ and the affine form of $\psi$ in $S$ that
$v^{*}$ satisfies a linear growth condition, and is therefore in
$\mathcal{A}_{t}$.

Let us consider
$(t,x,q,S)\in[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
and $v=(v_{s})_{s\in[t,T]}\in\mathcal{A}_{t}$. We now prove that

$\displaystyle\mathbb{E}\left[w\left(T,X^{t,x,S,v}_{T},q^{t,q,v}_{T},S^{t,S}_{T}\right)\right]\leqslant
w(t,x,q,S).$

We use the following notations for readability:

$\displaystyle\forall s\in[t,T],$ $\displaystyle\hskip
14.22636ptw(s,X^{t,x,S,v}_{s},q^{t,q,v}_{s},S^{t,S}_{s})=w^{t,x,q,S,v}_{s},$
$\displaystyle\forall s\in[t,T],$ $\displaystyle\hskip
14.22636pt\theta(s,q^{t,q,v}_{s},S^{t,S}_{s})=\theta^{t,q,S,v}_{s}.$

By Itô’s formula, we have $\forall s\in[\tau,T]$

$\displaystyle
dw^{t,x,q,S,v}_{s}=\mathcal{L}^{v}w^{t,x,q,S,v}_{s}ds+\left(\nabla_{S}w^{t,x,q,S,v}_{s}\right)^{\intercal}VdW_{s},$

where

$\displaystyle\mathcal{L}^{v}w^{t,x,q,S,v}_{s}=\ $
$\displaystyle\partial_{t}w^{t,x,q,S,v}_{s}-(v^{\intercal}S+v^{\intercal}\eta
v)\partial_{x}w^{t,x,q,S,v}_{s}+v^{\intercal}\nabla_{q}w^{t,x,q,S,v}_{s}$
$\displaystyle+(\overline{S}-S)^{\intercal}R^{\intercal}\nabla_{S}w^{t,x,q,S,v}_{s}+\frac{1}{2}\textrm{Tr}\left(\Sigma
D^{2}_{SS}w^{t,x,q,S,v}_{s}\right).$

From (12) and (18) we have

$\displaystyle\nabla_{S}w^{t,x,q,S,v}_{s}$ $\displaystyle=-\gamma
w^{t,x,q,S,v}_{s}\left(q^{t,q,v}_{s}+\nabla_{S}\theta^{t,q,S,v}_{s}\right)$
$\displaystyle=-\gamma
w^{t,x,q,S,v}_{s}\left(q^{t,q,v}_{s}+B(s)^{\intercal}q^{t,q,v}_{s}+2C(s)S^{t,S}_{s}+E(s)\right).$

We define $\forall s\in[t,T]$,

$\displaystyle\kappa^{q,S,v}_{s}$
$\displaystyle=-\gamma\left(q^{t,q,v}_{s}+B(s)^{\intercal}q^{t,q,v}_{s}+2C(s)S^{t,S}_{s}+E(s)\right),$
$\displaystyle\xi^{q,S,v}_{t,s}$
$\displaystyle=\exp\left(\int_{t}^{s}{\kappa^{q,S,v}_{\varrho}}^{\intercal}VdW_{\varrho}-\frac{1}{2}\int_{t}^{s}{\kappa^{q,S,v}_{\varrho}}^{\intercal}\Sigma\kappa^{q,S,v}_{\varrho}d\varrho\right).$

We then have

$\displaystyle
d\left(w^{t,x,q,S,v}_{s}\left(\xi^{q,S,v}_{t,s}\right)^{-1}\right)=\left(\xi^{q,S,v}_{t,s}\right)^{-1}\mathcal{L}^{v}w^{t,x,q,S,v}_{s}ds.$

By definition of $w$, $\mathcal{L}^{v}w^{t,x,q,S,v}_{s}\leqslant 0$. Moreover,
equality holds for the control reaching the supremum in (15). This supremum is
reached for the unique value

$\displaystyle v_{s}$
$\displaystyle=\frac{1}{2}\eta^{-1}\nabla_{q}\theta^{t,q,S,v}_{s}$
$\displaystyle=\frac{1}{2}\eta^{-1}\left(2A(s)q^{t,q,v}_{s}+B(s)S^{t,S}_{s}+D(s)\right),$

which corresponds to the case $(v_{s})_{s\in[t,T]}=(v^{*}_{s})_{s\in[t,T]}$.

As a consequence,
$\left(w^{t,x,q,S,v}_{s}\left(\xi^{q,S,v}_{t,s}\right)^{-1}\right)_{s\in[t,T]}$
is nonincreasing and therefore

$\displaystyle
w\left(T,X^{t,x,S,v}_{T},q^{t,q,v}_{T},S^{t,S}_{T}\right)\leqslant
w(t,x,q,S)\xi^{q,S,v}_{t,T},$

with equality when $(v_{s})_{s\in[t,T]}=(v^{*}_{s})_{s\in[t,T]}$.

Taking expectations we get

$\displaystyle\mathbb{E}\left[w\left(T,X^{t,x,S,v}_{T},q^{t,q,v}_{T},S^{t,S}_{T}\right)\right]\leqslant
w(t,x,q,S)\mathbb{E}\left[\xi^{q,S,v}_{t,T}\right].$

We proceed to prove that $\mathbb{E}\left[\xi^{q,S,v}_{t,T}\right]$ is equal
to 1. To do so, we use that $\xi^{q,S,v}_{t,t}=1$ and prove that
$(\xi^{q,S,v}_{t,s})_{s\in[t,T]}$ is a martingale under
$\left(\mathbb{P};\mathbb{F}=(\mathcal{F}_{s})_{s\in[t,T]}\right)$.

We know that $(q^{t,q,v}_{s})_{s\in[t,T]}$ satisfies a linear growth condition
with respect to $(S^{t,S}_{s})_{s\in[t,T]}$ since $v$ is an admissible
control. Given the form of $\kappa$, there exists a constant $C$ such that,
almost surely,

$\displaystyle\underset{s\in[t,T]}{\sup}{\parallel\kappa^{q,S,v}_{s}\parallel}^{2}\leqslant
C\left(1+\underset{s\in[t,T]}{\sup}{\parallel
W_{s}-W_{t}\parallel}^{2}\right).$

By using classical properties of the Brownian motion, we prove that

$\displaystyle\exists\epsilon>0,\forall s\in[t,T],\hskip
8.5359pt\mathbb{E}\left[\exp\left(\frac{1}{2}\int_{s}^{\left(s+\epsilon\right)\wedge
T}\left(\kappa^{q,S,v}_{\varrho}\right)^{\intercal}\Sigma\kappa^{q,S,v}_{\varrho}d\varrho\right)\right]<+\infty.$

Using a classical trick due to Beneš (see [29], Chapter 5), we see that
$(\xi^{q,S,v}_{t,s})_{s\in[t,T]}$ is a martingale under
$\left(\mathbb{P};\mathbb{F}=(\mathcal{F}_{s})_{s\in[t,T]}\right)$.

We obtain

$\displaystyle\mathbb{E}\left[w\left(T,X^{t,x,S,v}_{T},q^{t,q,v}_{T},S^{t,S}_{T}\right)\right]\leqslant
w(t,x,q,S),$

with equality when $(v_{s})_{s\in[t,T]}=(v^{*}_{s})_{s\in[t,T]}$.

We conclude that

$\displaystyle u\left(t,x,q,S\right)$
$\displaystyle=\underset{(v_{s})_{s\in[t,T]}\in\mathcal{A}_{t}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma\left(X^{t,x,S,v}_{T}+\left(q^{t,q,v}_{T}\right)^{\intercal}S^{t,S}_{T}-\ell\left(q^{t,q,v}_{T}\right)\right)\right)\right]$
$\displaystyle=\mathbb{E}\left[-\exp\left(-\gamma\left(X^{t,x,S,v^{*}}_{T}+\left(q^{t,q,v^{*}}_{T}\right)^{\intercal}S^{t,S}_{T}-\ell\left(q^{t,q,v^{*}}_{T}\right)\right)\right)\right]$
$\displaystyle=w\left(t,x,q,S\right).$

∎

We will next proceed to prove existence and uniqueness of a solution to the
system of ODEs (20) on $[0,T]$ with terminal condition (21), or equivalently
to (23) with terminal condition (24).161616The result in fact holds on
$(-\infty,T]$ as the initial time plays no role.

######  Theorem 2.

There exists a unique solution $A\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $B\in
C^{1}\left([0,T],\mathcal{M}_{d}(\mathbb{R})\right)$, $C\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $D\in
C^{1}\left([0,T],\mathbb{R}^{d}\right)$, $E\in
C^{1}\left([0,T],\mathbb{R}^{d}\right)$, $F\in
C^{1}\left([0,T],\mathbb{R}\right)$ to the system of ODEs (20) on $[0,T]$ with
terminal condition (21).

###### Proof.

To prove Theorem 2, it is enough, as explained in Remark 2, to show existence
and uniqueness for $A\in C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$,
$B\in C^{1}\left([0,T],\mathcal{M}_{d}(\mathbb{R})\right)$, and $C\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, or equivalently,
existence and uniqueness on $[0,T]$ of a solution $P\in
C^{1}\left([0,T],\mathcal{S}_{2d}(\mathbb{R})\right)$ to (23) with terminal
condition (24).

By Cauchy-Lipschitz theorem, there exists a unique maximal solution171717The
fact that $A$ and $C$ are symmetric is itself a consequence of Cauchy-
Lipschitz theorem since $(A,B,C)$ and $(A^{\intercal},B,C^{\intercal})$ are
solution of the same Cauchy problem. $(A,B,C)$ to the system of ODEs (22) with
terminal condition (21) defined on an open interval
$(t_{\text{min}},t_{\text{max}})\ni T$, and by Theorem 1, the associated
function $w$ defined by (12) corresponds to the value function of the problem
restricted to $[\tau,T]$ for all $\tau\in(t_{\text{min}},T)$.

To prove our result, we need to show that $t_{\text{min}}=-\infty$. For that
purpose, our strategy consists in proving that the matrix
$P(t)=\begin{pmatrix}A(t)&\frac{1}{2}B(t)\\\
\frac{1}{2}B(t)^{\intercal}&C(t)\end{pmatrix}$ cannot blow up in finite time.
This is proved by, first, finding (thanks to the control problem) lower and
upper bounds for the function $\theta$ which are, like $\theta$, polynomials
of degree at most $2$ in $(q,S)$, and then converting these bounds into bounds
for $P(t)$ in the sense of the natural order on symmetric matrices.181818For
$\underline{M},\overline{M}\in\mathcal{S}_{d}(\mathbb{R})$,
$\underline{M}\leq\overline{M}$ if and only if
$\overline{M}-\underline{M}\in\mathcal{S}^{+}_{d}(\mathbb{R})$.

By contradiction, let us assume that $t_{\text{min}}\in(-\infty,T)$ and let
$\tau\in(t_{\text{min}},T)$.

Let
$(t,x,q,S)\in[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$
and let us consider the sub-optimal strategy
$v=(0)_{s\in[t,T]}\in\mathcal{A}_{t}$ for which $\forall
s\in[t,T],q^{t,q,v}_{s}=q$ and

$\displaystyle\mathbb{E}\left[-\exp\left(-\gamma\left(X_{T}^{t,x,S,v}+(q^{t,q,v}_{T})^{\intercal}S^{t,S}_{T}-\ell(q^{t,q,v}_{T})\right)\right)\right]=\mathbb{E}\left[-\exp\left(-\gamma\left(x+q^{\intercal}S+q^{\intercal}\left(S^{t,S}_{T}-S\right)-q^{\intercal}\Gamma
q\right)\right)\right].$ (27)

Since $(S^{t,S}_{s})_{s\in[t,T]}$ follows multivariate Ornstein-Uhlenbeck
dynamics, we know that

$\displaystyle
S^{t,S}_{T}-S=\left(I-e^{-R\left(T-t\right)}\right)\left(\overline{S}-S\right)+\int_{t}^{T}e^{-R\left(T-s\right)}VdW_{s}.$
(28)

Then
$S^{t,S}_{T}-S\sim\mathcal{N}\left(\left(I-e^{-R\left(T-t\right)}\right)\left(\overline{S}-S\right),\Sigma_{t}\right)$,
where the covariance matrix is defined by

$\Sigma_{t}=\int_{t}^{T}e^{-R\left(T-s\right)}\Sigma
e^{-R^{\intercal}\left(T-s\right)}ds.\vspace{-0.2cm}$

Then,

$\displaystyle\mathbb{E}\left[-\exp\left(-\gamma\left(X_{T}^{t,x,S,v}+(q^{t,q,v}_{T})^{\intercal}S^{t,S}_{T}-\ell(q^{t,q,v}_{T})\right)\right)\right]$
$\displaystyle=\quad$
$\displaystyle-\exp\left(-\gamma(x+q^{\intercal}S)\right)\exp\left(-\gamma\left(q^{\intercal}\left(I-e^{-R\left(T-t\right)}\right)\left(\overline{S}-S\right)-q^{\intercal}\Gamma
q-\frac{1}{2}\gamma q^{\intercal}\Sigma_{t}q\right)\right).$

Since the strategy is sub-optimal, if we consider $\theta$ defined as in (18),
we have by Theorem 1

$\displaystyle-\exp\left(-\gamma\left(x+q^{\intercal}S+\theta(t,q,S)\right)\right)$
$\displaystyle\geq-\exp\left(-\gamma(x+q^{\intercal}S)\right)\exp\left(-\gamma\left(q^{\intercal}\left(I-e^{-R\left(T-t\right)}\right)\left(\overline{S}-S\right)-q^{\intercal}\Gamma
q-\frac{1}{2}\gamma q^{\intercal}\Sigma_{t}q\right)\right).$ (29)

We conclude that for all
$(t,q,S)\in[\tau,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$,

$\displaystyle\theta(t,q,S)=\begin{pmatrix}q\\\
S\end{pmatrix}^{\intercal}P(t)\begin{pmatrix}q\\\
S\end{pmatrix}+\begin{pmatrix}D(t)\\\
E(t)\end{pmatrix}^{\intercal}\begin{pmatrix}q\\\ S\end{pmatrix}+F(t)$
$\displaystyle\geq\quad$ $\displaystyle\begin{pmatrix}q\\\
S\end{pmatrix}^{\intercal}\begin{pmatrix}-\frac{\gamma}{2}\Sigma_{t}-\Gamma&-\frac{1}{2}\left(I-e^{-R\left(T-t\right)}\right)\\\
-\frac{1}{2}\left(I-e^{-R^{\intercal}\left(T-t\right)}\right)&0\end{pmatrix}\begin{pmatrix}q\\\
S\end{pmatrix}+\overline{S}^{\intercal}\left(I-e^{-R^{\intercal}\left(T-t\right)}\right)q.$

We therefore necessarily have, for the natural order on symmetric matrices,

$\forall t\in[\tau,T],\quad
P(t)\geq\begin{pmatrix}-\frac{\gamma}{2}\Sigma_{t}-\Gamma&-\frac{1}{2}\left(I-e^{-R\left(T-t\right)}\right)\\\
-\frac{1}{2}\left(I-e^{-R^{\intercal}\left(T-t\right)}\right)&0\end{pmatrix}.$

Now, for
$(t,x,q,S)\in[\tau,T]\times\mathbb{R}\times\mathbb{R}^{d}\times\mathbb{R}^{d}$,
we have

$\displaystyle\underset{v\in\mathcal{A}_{t}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma\left(X^{t,x,S,v}_{T}+(q^{t,q,v}_{T})^{\intercal}S^{t,S}_{T}-(q^{t,q,v}_{T})^{\intercal}\Gamma
q^{t,q,v}_{T}\right)\right)\right]$ $\displaystyle=$
$\displaystyle\underset{v\in\mathcal{A}_{t}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma\left(x+q^{\intercal}S+\int_{t}^{T}(q^{t,q,v}_{s})^{\intercal}dS_{s}-\int_{t}^{T}L(v_{s})ds-(q^{t,q,v}_{T})^{\intercal}\Gamma
q^{t,q,v}_{T}\right)\right)\right]$ $\displaystyle\leq$
$\displaystyle\exp\left(-\gamma\left(x+q^{\intercal}S\right)\right)\underset{v\in\mathcal{A}_{t}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma\left(\int_{t}^{T}(q^{t,q,v}_{s})^{\intercal}dS_{s}\right)\right)\right],$

If $(v_{s})_{s\in[t,T]}\in\mathcal{A}_{t}$, it is straightforward to see that
the process $(q^{t,q,v}_{s})_{s\in[t,T]}$ is in the space of admissible
controls $\mathcal{A}_{t}^{Merton}$ defined by (35) in Appendix B (in which we
study a Merton problem that can be regarded as a limit case of ours when the
execution costs and terminal costs vanish). Therefore,

$\displaystyle\underset{v\in\mathcal{A}_{t}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma\left(X^{t,x,S,v}_{T}+(q^{t,q,v}_{T})^{\intercal}S^{t,S}_{T}-(q^{t,q,v}_{T})^{\intercal}\Gamma
q^{t,q,v}_{T}\right)\right)\right]$ $\displaystyle\leq$
$\displaystyle\exp\left(-\gamma\left(x+q^{\intercal}S\right)\right)\underset{(q_{s})_{s\in[t,T]}\in\mathcal{A}_{t}^{Merton}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma\left(\int_{t}^{T}q^{\intercal}_{s}dS_{s}\right)\right)\right].$
(30)

As shown in Appendix B, inequality (2.3) writes

$\displaystyle-\exp\left(-\gamma\left(x+q^{\intercal}S+\theta(t,q,S)\right)\right)\leq-\exp\left(-\gamma\left(x+q^{\intercal}S+\hat{\theta}(t,S)\right)\right),$

where
$\hat{\theta}(t,S)=S^{\intercal}\hat{C}(t)S+\hat{E}(t)^{\intercal}S+\hat{F}(t)$
with $\hat{C}\in C^{1}\left([\tau,T],\mathcal{S}_{d}(\mathbb{R})\right)$,
$\hat{E}\in C^{1}\left([\tau,T],\mathbb{R}^{d}\right)$, $\hat{F}\in
C^{1}\left([\tau,T],\mathbb{R}\right)$ defined by

$\begin{cases}\hat{C}(t)=\frac{1}{2\gamma}(T-t)R^{\intercal}\Sigma^{-1}R,\\\
\hat{E}(t)=-\frac{1}{\gamma}(T-t)R^{\intercal}\Sigma^{-1}R\overline{S},\\\
\hat{F}(t)=\frac{1}{4\gamma}(T-t)^{2}\text{Tr}\left(R^{\intercal}\Sigma^{-1}R\Sigma\right)+\frac{1}{2\gamma}(T-t)\overline{S}^{\intercal}R^{\intercal}\Sigma^{-1}R.\end{cases}$

We conclude that for all
$(t,q,S)\in[\tau,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}$,

$\displaystyle\theta(t,q,S)=\begin{pmatrix}q\\\
S\end{pmatrix}^{\intercal}P(t)\begin{pmatrix}q\\\
S\end{pmatrix}+\begin{pmatrix}D(t)\\\
E(t)\end{pmatrix}^{\intercal}\begin{pmatrix}q\\\ S\end{pmatrix}+F(t)$
$\displaystyle\leq\quad$ $\displaystyle\begin{pmatrix}q\\\
S\end{pmatrix}^{\intercal}\begin{pmatrix}0&0\\\
0&\hat{C}(t)\end{pmatrix}\begin{pmatrix}q\\\
S\end{pmatrix}+\begin{pmatrix}0\\\
\hat{E}(t)\end{pmatrix}^{\intercal}\begin{pmatrix}q\\\
S\end{pmatrix}+\hat{F}(t).$

Therefore,

$\forall t\in[\tau,T],\quad P\left(t\right)\leq\begin{pmatrix}0&0\\\
0&\hat{C}(t)\end{pmatrix}=\begin{pmatrix}0&0\\\
0&\frac{1}{2\gamma}(T-t)R^{\intercal}\Sigma^{-1}R\end{pmatrix}.$

We have therefore $\forall\tau\in(t_{\text{min}},T)$, $\forall t\in[\tau,T]$:

$\begin{pmatrix}-\frac{\gamma}{2}\Sigma_{t}-\Gamma&-\frac{1}{2}\left(I-e^{-R\left(T-t\right)}\right)\\\
-\frac{1}{2}\left(I-e^{-R^{\intercal}\left(T-t\right)}\right)&0\end{pmatrix}\leq
P\left(t\right)\leq\begin{pmatrix}0&0\\\
0&\frac{1}{2\gamma}(T-t)R^{\intercal}\Sigma^{-1}R\end{pmatrix}.$

As $t_{\text{min}}$ is supposed to be finite, there exists
$\underline{M},\overline{M}\in\mathcal{S}_{d}(\mathbb{R})$ with
$\underline{M}\leq\overline{M}$ such that $\forall t\in[t_{\text{min}},T]$,
$P(t)$ stays in the compact set
$\\{M\in\mathcal{S}_{d}(\mathbb{R})\,\left|\underline{M}\leq
M\leq\overline{M}\right.\\}$. This contradicts the maximality of the solution,
hence $t_{\text{min}}=-\infty$.
∎

Theorem 2 implies that Theorem 1 can be applied with $\tau=0$. In particular,
our optimal execution problem is solved and the optimal strategy is given by
the closed-loop feedback control (26). In the next section, we illustrate our
results with simulations of prices and numerical approximations of the optimal
strategies.

##  3 Numerical results

In this section, we present several applications of our results. We first
exemplify the use of the optimal strategy derived in the above section by a
trader wishing to unwind a single-asset portfolio. In particular, we show that
the optimal liquidation strategy in our model with mean reversion is really
different from that derived in the Almgren-Chriss model. We also demonstrate
the usefulness of our results for statistical arbitrage purposes in the one-
asset case. The one-asset examples are based on data from the foreign exchange
(FX) market. We then illustrate our results in the multi-asset case by
considering a pair of two cointegrated French stocks. We start with a two-
asset portfolio liquidation problem and compare the optimal liquidation
strategy in our model with that obtained in a multi-asset Almgren-Chriss
model. We then illustrate the use of our results for statistical arbitrage
purposes (a pair trading strategy in our case).191919Throughout this section,
optimal trading strategies are computed by approximating the solution of the
Riccati ODEs using implicit Euler schemes.

In the Almgren-Chriss model used for carrying out comparisons, the price
dynamics is of the form $dS_{t}=V_{\textrm{AC}}dW_{t}$, where
$V_{\textrm{AC}}\in\mathcal{M}_{d,k}(\mathbb{R})$, i.e. a simple Bachelier
dynamics (with correlations). This dynamics differs from that of the OU model
we use throughout the paper (i.e. $dS_{t}=R(\overline{S}-S_{t})dt+VdW_{t}$)
when $R\neq 0$. In particular, if prices exhibit mean reversion or a
cointegrated behavior, as is the case in our examples, the classical Almgren-
Chriss model shall not properly take the true multivariate dynamics of prices
into account, with sometimes important consequences in terms of risk
management.

###  3.1 Single-asset case

In order to illustrate the use of the optimal strategies we derived in the
above section, let us start with a single-asset case. For that purpose, we use
data from the FX market, in which asset prices often exhibit mean reversion.
More precisely, we consider a FX futures contract (hereafter CDU1) on the
currency pair Canadian Dollar (CAD) / US Dollar (USD) that is exchanged on the
Chicago Mercantile Exchange. The contract specifications are given in Table 1.

Underlying asset | Canadian Dollar
---|---
Quotation currency | US Dollar
Contract size | CAD 100000
Expiry date | September 14, 2021
Table 1: CDU1 contract specifications

We plot in Figure 1 the mid-price of CDU1,202020CDU1 is usually quoted in USD
cents per CAD. However, to use our model, the price must take account of the
contract size and be the contract value in USD. sampled every 60 seconds
during the regular trading hours ($02$:$00$-$16$:$00$ Central
Time),212121Although CDU1 is quoted continuously with a 60-minute break each
day beginning at $16$:$00$ CT, we only consider the trading hours between
$02$:$00$ and $16$:$00$ CT because the contract is only liquid during these
hours corresponding to European and American market activity. over the three
following trading days: August 11, August 12, and August 13, 2021.

Figure 1: Mid-price of CDU1 sampled every 60 seconds during the regular
trading hours ($02$:$00$-$16$:$00$ CT). Top left: August 11, 2021. Top right:
August 12, 2021. Bottom: August 13, 2021.

####  3.1.1 Liquidation problem

We consider the case of a trader wishing to unwind a long position in 2250
contracts222222This represents roughly $5\%$ of the average daily traded
volume over the period considered in this example. during the third day, i.e.
on August 13, 2021.

To exemplify the use of our strategies, we first estimate Ornstein-Uhlenbeck
parameters using prices from the two preceding trading days: August 11 and
August 12, 2021. Coefficients are classically estimated using least squares
regression.232323A time discretization of an Ornstein-Uhlenbeck model gives
rise to an Auto-Regressive model of order $1$, or $\textrm{AR}(1)$. The
parameters of an $\textrm{AR}(1)$ model are classically estimated by using
least squares regression. Conversion of $\textrm{AR}(1)$ coefficients into
their continuous-time counterparts is straightforward. In order to set the
value of the execution cost / temporary market impact parameter $\eta$, we use
a similar argument as in [7]: we suppose that the additional cost incurred per
contract when trading a given volume is proportional to the participation rate
to the market. More precisely, for each percent of participation rate (in
practice we consider a flat volume curve that matches the average daily
volume), a cost corresponding to half the bid-ask spread242424The average bid-
ask spread is close to the tick value equal to $\$5$ per contract. is
incurred. Rounding values, this results in setting $\eta=5\cdot 10^{-3}\
\$\cdot\textrm{day}$. For the terminal penalty parameter $\Gamma$, we set a
high value to enforce complete liquidation by the end of the trading day. For
the risk aversion parameter $\gamma$, we choose an intermediate value that
does not neutralize any of the financial effects our model could illustrate.
Choosing a too high value of the risk aversion parameter would force the
trader to liquidate quickly with no illustration of the impact of mean
reversion. On the contrary, choosing a too low value of the risk aversion
parameter would result in the trader accumulating unrealistically large
positions to benefit from mean reversion at the expense of the original
liquidation problem.

The resulting values used to run our algorithms are given in Table 2.252525In
the one-asset case, $\Sigma=VV^{\intercal}$ is a scalar. We classically write
it as $\sigma^{2}$ and document the value of $\sigma$.

Parameter | Value
---|---
$T$ | $1\ \textrm{day}$
$q_{0}$ | $2250$
$S_{0}$ | $\$79835$
$R$ | $5.1\ \textrm{day}^{-1}$
$\overline{S}$ | $\$79887$
$\sigma$ | $243.67\ \$\cdot\textrm{day}^{-\frac{1}{2}}$
$\eta$ | $5\cdot 10^{-3}\ \$\cdot\textrm{day}$
$\Gamma$ | $\$100\ $
$\gamma$ | $2\cdot 10^{-5}\ \textrm{\$}^{-1}$
Table 2: Value of the parameters.

We plot in Figure 2 the asset price trajectory $(S_{t})_{t\in[0,T]}$ on August
13, 2021 and the inventory process $(q_{t})_{t\in[0,T]}$ corresponding to the
use of the optimal liquidation strategy derived in the previous
section.262626In what follows, this strategy is often referred to as ACOU
(Almgren-Chriss under Ornstein-Uhlenbeck dynamics) strategy. For comparison
purposes, we also plot the inventory process when using a classical Almgren-
Chriss (AC) strategy.272727To compute the AC strategy, we estimate the
parameter $V_{\textrm{AC}}$ of the Bachelier dynamics. This parameter is a
scalar in our one-asset case and we denote it by $\sigma_{\textrm{AC}}$
instead of $V_{\textrm{AC}}$. A simple estimation based on price increments
leads to $\sigma_{\textrm{AC}}=244.02\ \$\cdot\textrm{day}^{-\frac{1}{2}}$
which slighlty differs from $\sigma$ because the drift term in the OU model
captures part of the variance.

The results shown in Figure 2 deserve several remarks. First, the optimal
liquidation strategy in our model with mean reversion is different from that
derived in the Almgren-Chriss model. In particular, the liquidation process is
significantly faster in the latter case because the process of unwinding the
portfolio appears far riskier to a trader who believes that the price evolves
as a Brownian motion than to another one who believes in a mean-reverting
Ornstein-Uhlenbeck dynamics. Second, in the case of the ACOU strategy, the
trader progressively unwinds her long position over the trading day but also
takes advantage of mean reversion. When the price is below $\overline{S}$, the
trader tends to reduce the pace of her selling process or even buys some
contracts. Symmetrically, when the price is above $\overline{S}$, the trader
tends to sell at a faster pace. One exception to the above should nevertheless
be noticed, close to time $T$. Indeed, because of the high value of the final
penalty, close to time $T$ the trader focuses more on liquidating her
portfolio and cares less about price oscillations. In particular, she buys
back some contracts to end up flat because she previously went short to bet on
the reversion of the price towards $\overline{S}$.

Figure 2: Top: CDU1 price trajectory on August 13, 2021 –
$(S_{t})_{t\in[0,T]}$. Middle: Trajectory of the inventory when using the
optimal strategy corresponding to the estimated Ornstein-Uhlenbeck process –
$(q_{t})_{t\in[0,T]}$. Bottom: Trajectory of the inventory when using the
optimal strategy corresponding to a Brownian motion (Bachelier) model for the
price (classical Almgren-Chriss strategy).

####  3.1.2 Impact of mean reversion

We have just seen that mean reversion plays a key role in the characteristics
of the strategy. In order to further study the impact of the mean-reversion
parameter $R$, we consider the same parameters as in Table 2 except that we
force the mean-reversion parameter $R$ to have the following values: $R=0\
\textrm{day}^{-1}$, $R=3\ \textrm{day}^{-1}$, and $R=10\ \textrm{day}^{-1}$.
Moreover, unlike what we did previously, we consider now a price trajectory in
line with the choice of the parameter $R$. For that purpose, we simulate price
trajectories by using the same path of the Brownian motion but different
values of $R$.

In Figure 3, we plot an instance of the (simulated) trajectories of the price
process $(S_{t})_{t\in[0,T]}$ for the different values of $R$ and the
corresponding inventory processes $(q_{t})_{t\in[0,T]}$ – along with the
associated trading volume (or trading speed) processes $(v_{t})_{t\in[0,T]}$ –
when using the optimal execution strategy.

Figure 3: Top: Simulated trajectories of the asset price for the different
values of $R$ – $(S_{t})_{t\in[0,T]}$. Middle: Corresponding inventory
processes when using the optimal execution strategy – $(q_{t})_{t\in[0,T]}$.
Bottom: Trajectories of the corresponding trading volume (or trading speed)
processes – $(v_{t})_{t\in[0,T]}$.

The results of Figure 3 confirm that the way the optimal strategy handles risk
depends strongly on the mean-reversion parameter $R$. In particular, when $R$
is large, the trader acts almost282828Close to time $T$, because of the high
value of $\Gamma$, the trader focuses more on unwinding her portfolio and
cares less about price oscillations, as can clearly be seen in Figure 3. as if
she was performing a VWAP/TWAP292929VWAP and TWAP respectively mean Volume-
Weighted Average Price and Time-Weighted Average Price. VWAP and TWAP
strategies are commonly used by traders to execute orders at a price as close
as possible to the average price of all transactions over a given period.
execution strategy plus a mean-reverting statistical arbitrage strategy. In
particular, in the case where $R=10\ \textrm{day}^{-1}$, the process
$(v_{t})_{t\in[0,T]}$ oscillates around its average (which is of course fixed
by the total number of contracts to sell). These oscillations are highly
correlated with those of $(S_{t})_{t\in[0,T]}$: the trader sells faster when
the price is above $\overline{S}$ and slower (she even buys sometimes) when it
is below $\overline{S}$.

####  3.1.3 Statistical arbitrage

Given the observations of the previous subsection, it is natural to illustrate
how our model can be used to build a statistical arbitrage strategy. For that
purpose, we consider a trader with no initial inventory who starts trading the
futures contract CDU1 on August 13, 2021 and wants to maximize the expected
utility of her PnL at the end of the day (with no final penalty).

To run our algorithm we use the same parameters as in Table 2 except that
$\Gamma=0$ and $q_{0}=0$. The results are plotted in Figure 4: the price
process $(S_{t})_{t\in[0,T]}$, the inventory process $(q_{t})_{t\in[0,T]}$
when using the optimal strategy, and the associated trajectory of the PnL,
i.e. the process $(X_{t}+q_{t}S_{t})_{t\in[0,T]}$.

Figure 4: Top: CDU1 price trajectory on August 13, 2021 –
$(S_{t})_{t\in[0,T]}$. Middle: Trajectory of the inventory when using the
optimal strategy starting from $q_{0}=0$ – $(q_{t})_{t\in[0,T]}$. Bottom:
Corresponding trajectory of the PnL – $(X_{t}+q_{t}S_{t})_{t\in[0,T]}$.

Because the trader believes that the price mean reverts around $\overline{S}$,
her optimal strategy, in the absence of execution costs, would consist in
having a long position when the price is below $\overline{S}$ and a short
position when the price is above $\overline{S}$. The optimal strategy when
execution costs are taken into account is however more complex because the
control of execution costs introduces inertia in the position of the trader.
It consists instead in trading progressively to target a long position when
the price is below $\overline{S}$ and a short position when the price is above
$\overline{S}$. However, because of inertia, it happens that the position
remains long while the price is still far above $\overline{S}$, as exemplified
in Figure 4. We see nevertheless that the strategy would have been profitable
to the trader.

Another interesting experiment for assessing the performance of the strategy,
when used for pure statistical arbitrage, consists in testing it on simulated
price trajectories (using the same Ornstein-Uhlenbeck parameters as above for
both simulating prices and computing the optimal strategies). We plot in
Figure 5, the distribution of the final PnL after $1500$ simulations of the
price process. We see that our strategy allows to make money by taking
advantage of the mean reversion: we get a positive final profit of $\$107698$
on average, with a standard deviation of $\$57791$, and the distribution looks
skewed to the right (towards profits rather than losses).

Figure 5: Distribution of the final PnL for the statistical arbitrage strategy
on CDU1 (for $1500$ simulations).

###  3.2 Multi-asset case

We now come to the use of our optimal strategies in the multi-asset case when
asset prices exhibit a cointegrated behaviour that can be modeled by a multi-
OU process. For that purpose, we use data from two French stocks within the
banking sector: BNP Paribas (hereafter BNP) and Société Générale (hereafter
GLE).

We plot in Figure 6 the mid-prices of BNP and GLE sampled every 60 seconds
during the regular trading hours ($09$:$00$-$17$:$30$) over the week August
09-August 13, 2021. We clearly see that the stock prices of the two companies
are driven by the same factors and should be cointegrated.303030The existence
of a cointegration vector is confirmed by a Johansen’s cointegration test (see
below).

Figure 6: Mid-prices of BNP (left axis) and GLE (right axis) sampled every 60
seconds during the regular trading hours ($09$:$00$-$17$:$30$) over the week
August 09-August 13, 2021.

####  3.2.1 Portfolio liquidation in the presence of cointegration

We consider the case of a trader wishing to unwind a portfolio with $75000$
shares of BNP and $75000$ shares of GLE on August 13, 2021.313131This
represents roughly $5\%$ of the average daily traded volume over the period
considered in this example.

Similarly to our one-asset example, we consider that the trader estimates the
parameters of a multi-OU model using prices from the four preceding trading
days, here August 9, 10, 11, and 12, 2021. These parameters are estimated
using classical linear regression techniques.323232A time discretization of a
multi-OU model gives rise to a Vector Auto-Regressive model of order $1$, or
$\textrm{VAR}(1)$. The parameters of a $\textrm{VAR}(1)$ model are classically
estimated by using least squares regression. Conversion of $\textrm{VAR}(1)$
coefficients into their continuous-time multi-OU counterparts is
straightforward. Referring to BNP and GLE by respectively using the
superscripts $1$ and $2$, the estimated values of the parameters are given in
Table 3.

Parameter | Estimate
---|---
$R$ | $\begin{pmatrix}0.33&3.95\\\ -2.52&10.23\end{pmatrix}\ \textrm{day}^{-1}$
$\overline{S}$ | $\left(\overline{S}^{1},\overline{S}^{2}\right)=\left(\textrm{€}54.23,\textrm{€}27.45\right)$
$\Sigma$ | $\begin{pmatrix}0.47&0.20\\\ 0.20&0.14\end{pmatrix}\textrm{€}^{2}\cdot\textrm{day}^{-1}$
Table 3: Multi-OU estimated parameters for the pair (BNP, GLE).

The use of a Johansen’s cointegration test333333We use the Trace test and not
the Maximum Eigenvalue test in what follows. rejects the hypothesis of no
cointegration but does not reject a cointegration rank $r=1$ for the pair
(BNP, GLE).343434Johansen’s approach is based on the VAR($1$) formulation
$\Delta S_{t}=a+\Pi S_{t-1}+\epsilon_{t}$, where $a\in\mathbb{R}^{d}$,
$\Pi\in\mathcal{M}_{d}(\mathbb{R})$ and $\epsilon$ is normally distributed. In
a nutshell, it iteratively tests the null hypothesis $\textrm{rank}(\Pi)\leq
k$ (corresponding to the existence of at most $k$ linearly independent
cointegration vectors) for different values of $k$ using likelihood ratio
statistics that follow tabulated distributions (see [28] for more details). In
practice, the cointegration rank retained is the first value of $k$ for which
the null hypothesis is not rejected. Furthermore, the estimated value of the
matrix $R$ suggests that the space of cointegration vectors is spanned by
$(1,-3.46)$.

We give in Table 4 the detailed results for the Johansen’s cointegration Trace
test.

Null Hypothesis | Trace statistics | Critical Value | Conclusion
---|---|---|---
$r\leq 0$ | 16.77 | 15.49 | Rejected
$r\leq 1$ | 2.293 | 3.841 | Not rejected
Table 4: Johansen’s cointegration Trace test results (critical values are
given for a significance level of $95\%$).

To exemplify the use of our strategies and illustrate the different effects in
the multi-asset case, we run our algorithms for two different values of the
risk aversion parameter $\gamma$. More precisely, we consider the parameters
stated in Table 5 (values of the multi-OU parameters are not recalled, see
Table 3).353535The same logic as in the one-asset case has been applied for
the choice of the parameters.

Parameter | Value
---|---
$T$ | $1\ \textrm{day}$
$q_{0}$ | $(75000,75000)$
$S_{0}$ | $\left(S_{0}^{1},S_{0}^{2}\right)=\left(\textrm{€}54.4,\textrm{€}27.48\right)$
$\eta$ | $\begin{pmatrix}4\cdot 10^{-7}&0\\\ 0&2\cdot 10^{-7}\end{pmatrix}\ \textrm{€}\cdot\textrm{day}$
$\Gamma$ | $\textrm{€}100\times I_{2}$
$\gamma$ |  $2\cdot 10^{-5}\ \textrm{€}^{-1}$ or $2\cdot 10^{-3}\ \textrm{€}^{-1}$
Table 5: Value of the parameters.

We plot in Figures 7 and 8 the trajectory on August 13, 2021 of the price
process $(S_{t})_{t\in[0,T]}$ and the spread process
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$
corresponding to the cointegration vector, along with the inventory process
$(q_{t})_{t\in[0,T]}$ corresponding to the use of the optimal liquidation
strategy with $\gamma=2\cdot 10^{-5}\ \textrm{€}^{-1}$ and $\gamma=2\cdot
10^{-3}\ \textrm{€}^{-1}$ respectively.363636As above, this strategy is
referred to as ACOU (Almgren-Chriss under multivariate Ornstein-Uhlenbeck
dynamics) strategy. For comparison purposes, we also plot the inventory
process when using a classical Almgren-Chriss (AC) strategy.373737To compute
the AC strategy, we estimate the parameter $V_{\textrm{AC}}$ of the Bachelier
dynamics. A simple estimation based on price increments leads to
$\Sigma_{\textrm{AC}}=V_{\textrm{AC}}{V_{\textrm{AC}}}^{\intercal}=\begin{pmatrix}0.48&0.19\\\
0.19&0.13\end{pmatrix}\ \textrm{€}^{2}\cdot\textrm{day}^{-1}$ which slighlty
differs from $\Sigma$ because the drift term in the OU model captures part of
the variance.

Figure 7: Top two plots: BNP and GLE price trajectories on August 13, 2021 –
$(S_{t})_{t\in[0,T]}$ – and trajectory of the spread corresponding to the
cointegration vector on August 13, 2021 –
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$.
Bottom two plots: Trajectory of the inventories when using the optimal
strategy corresponding to the estimated multi-OU process with $\gamma=2\cdot
10^{-5}\ \textrm{€}^{-1}$ – $(q_{t})_{t\in[0,T]}$ – and when using the optimal
strategy corresponding to correlated Brownian motions for the stock prices
(classical Almgren-Chriss strategy with $\gamma=2\cdot 10^{-5}\
\textrm{€}^{-1}$). Figure 8: Top two plots: BNP and GLE price trajectories on
August 13, 2021 – $(S_{t})_{t\in[0,T]}$ –, and trajectory of the spread
corresponding to the cointegration vector on August 13, 2021 –
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$.
Bottom two plots: Trajectory of the inventories when using the optimal
strategy corresponding to the estimated multi-OU process with $\gamma=2\cdot
10^{-3}\ \textrm{€}^{-1}$ – $(q_{t})_{t\in[0,T]}$ –, and when using the
optimal strategy corresponding to correlated Brownian motions for the stock
prices (classical Almgren-Chriss strategy with $\gamma=2\cdot 10^{-3}\
\textrm{€}^{-1}$).

We clearly see, for both values of $\gamma$ that the ACOU strategy is
different from the AC one383838It is noteworthy that the trading curves for
BNP and GLE are almost the same in the AC case. Because of the value of
$S_{0}$ and $\eta$ this is unsurprising. although they both succeed in
unwinding the portfolio. This is easily understandable: because of the
presence of the matrix $R$ in the multi-OU model, there is a drift in the
dynamics of the prices that can be exploited to make money (while still
controlling market risk).

It is interesting to understand the difference between what happens when
$\gamma$ is small versus what happens when $\gamma$ is large because the
trading curves exhibit very different properties. When $\gamma$ is small, we
observe that the trader oversells the two stocks (before buying back close to
time $T$ to unwind the portfolio) and speculates therefore on the reversion of
the two stock prices towards $\overline{S}^{1}$ and $\overline{S}^{2}$
respectively – because prices are above these values. On the contrary, when
$\gamma$ is large, shorting the two stocks simultaneously appears too risky
and, once the portfolio has been partially liquidated, the trader uses instead
a long/short strategy. In that case, the trader seems in fact to apply a
statistical arbitrage strategy related to the spread process
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$:
for $t\geq 0.35$, when the spread process is below $0$, the trader is long BNP
and short GLE – she buys the spread – while it becomes (slighlty) long GLE and
(slighlty) short BNP – she shorts the spread – at the very end of the period
when the spread becomes positive. The fact that the trader “trades the spread”
will appear even more clearly in what follows as we focus on statistical
arbitrage strategies.

####  3.2.2 Statistical arbitrage

Given the previous remarks, it is natural to illustrate the use of our model
in the context of pure statistical arbitrage. For that purpose, we consider a
trader with no initial inventory who starts trading on August 13, 2021 and
wants to maximize the expected utility of her PnL at the end of the day (with
no final penalty).

Figure 9: Top: Trajectory of the spread on August 13, 2021 –
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$.
Middle: Trajectory of the inventories when using the optimal strategy
corresponding to the estimated multi-OU process with $\gamma=2\cdot 10^{-3}\
\textrm{€}^{-1}$ – $(q_{t})_{t\in[0,T]}$. Bottom: Trajectory of the PnL –
$(X_{t}+q^{\intercal}_{t}S_{t})_{t\in[0,T]}$.

To run our algorithm we use the same parameters as in Tables 3 and 5 with
$\gamma=2\cdot 10^{-3}\ \textrm{€}^{-1}$ except that $\Gamma=0$ and
$q_{0}=(0,0)$. The results are plotted in Figure 9: the spread process
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$,
the inventory process $(q_{t})_{t\in[0,T]}$ when using the optimal strategy,
and the associated trajectory of the PnL, i.e. the process
$(X_{t}+q^{\intercal}_{t}S_{t})_{t\in[0,T]}$.

We clearly see that for $\gamma=2\cdot 10^{-3}\ \textrm{€}^{-1}$ the optimal
strategy is a long/short strategy. As can be seen in Figure 9 (middle plot),
the process $(-3.46q^{1}_{t})_{t\in[0,T]}$ appears to be in line with
$(q^{2}_{t})_{t\in[0,T]}$. This confirms that the strategy consists mainly in
“buying or selling the spread” depending on the sign of the spread process
$((S^{1}_{t}-\overline{S}^{1})-3.46(S^{2}_{t}-\overline{S}^{2}))_{t\in[0,T]}$.

Finally, as in the previous subsection, we test our optimal strategy on
simulated price trajectories (using the same multi-OU parameters as above for
both simulating prices and computing the optimal strategies). We plot in
Figure 10, the distribution of the final PnL after $1500$ simulations of the
price process when using the optimal strategy with the parameters of Table 5
(with $\gamma=2\cdot 10^{-3}\ \textrm{€}^{-1}$) except that $\Gamma=0$ and
$q_{0}=(0,0)$ (because we focus on statistical arbitrage). We see that our
strategy allows to make money by taking advantage of the price dynamics: we
get a positive final profit of $\textrm{€}2230$ on average, with a standard
deviation of $\textrm{€}1145$, and the distribution is, as above, skewed
towards profits rather than losses.393939The PnLs are smaller in absolute
value than in the one-asset example because the value of $\gamma$ is higher
here.

Figure 10: Distribution of the final PnL for the statistical arbitrage
strategy on BNP and GLE (for $1500$ simulations).

## Conclusion

In this paper, we have shown how to account for cross-asset co-movements when
executing trades in multiple assets. In our model, the agent has an
exponential utility and the prices have multivariate Ornstein-Uhlenbeck
dynamics, capturing the complex cross-asset dynamics of prices better than
correlated Brownian motions only. The advantage of our approach is twofold:
(i) it better accounts for risk at the portfolio level, and (ii) it is
versatile and can be used for basket execution and statistical arbitrage.

The advantages for practitioners are numerous. Considering asset execution
within a portfolio allows to manage risk across a wider basket of assets
rather than considering only the risk of a single trade. Agents can hold
securities on their balance sheets for longer, reducing market impact and
execution costs. Moreover, from a regulation point of view, multivariate
optimal execution models that naturally offset risks in a portfolio are of
great interest. In fact, the new FRTB (Fundamental Review of the Trading Book)
regulation will lead practitioners to assess liquidity risks within a
centralized risk book for capital requirements. In this context, our model can
reduce the liquidity risk of the execution process by taking into account the
joint dynamics of the assets.

##  Appendix A Appendix – Multi-asset optimal execution with correlated
Brownian motions and execution costs

We consider in this appendix the problem of multi-asset optimal execution in
the case where prices are correlated arithmetic Brownian motions. This problem
is a special case of that presented in this paper, corresponding to $R=0$ in
the dynamics (2) of the asset prices. Therefore, the results presented in the
paper apply. However, when $R=0$, as mentioned in Remark 2, the system of ODEs
(20) simplifies since a trivial solution to the last five equations is
$B=C=D=E=F=0$. Therefore, the problem boils down to finding $A\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$ solution of the following
terminal value problem:

$\displaystyle\begin{cases}A^{\prime}(t)&=\frac{\gamma}{2}\Sigma-A(t)\eta^{-1}A(t)\\\
A(T)&=-\Gamma.\end{cases}$ (31)

In this appendix we show that, when $\Sigma\in S^{++}_{d}(\mathbb{R})$, $A$
can be found in closed form.

For that purpose, we introduce the change of variables

$a(t)=\eta^{-\frac{1}{2}}A(t)\eta^{-\frac{1}{2}}\quad\forall t\in[0,T]$

and notice that (31) is equivalent to the terminal value problem

$\displaystyle\begin{cases}a^{\prime}(t)=&\hat{A}^{2}-a(t)^{2}\\\
a(T)=&-C,\end{cases}$ (32)

where
$\hat{A}=\sqrt{\frac{\gamma}{2}}\left(\eta^{-\frac{1}{2}}\Sigma\eta^{-\frac{1}{2}}\right)^{\frac{1}{2}}\in\mathcal{S}^{++}_{d}(\mathbb{R})$
and
$C=\eta^{-\frac{1}{2}}\Gamma\eta^{-\frac{1}{2}}\in\mathcal{S}^{+}_{d}(\mathbb{R})$.

To solve (32) we use a classical trick for Riccati equations in the following
Proposition:

######  Proposition 3.

Let $\xi:[0,T]\rightarrow\mathcal{S}_{d}(\mathbb{R})$ defined as

$\displaystyle\xi\left(t\right)=-\frac{\hat{A}^{-1}}{2}\left(I-e^{-2\hat{A}\left(T-t\right)}\right)-e^{-\hat{A}\left(T-t\right)}\left(C+\hat{A}\right)^{-1}e^{-\hat{A}\left(T-t\right)}$
(33)

be the unique solution of the linear ODE

$\displaystyle\begin{cases}\xi^{\prime}(t)=\hat{A}\xi(t)+\xi(t)\hat{A}+I_{d}\\\
\xi(T)=-\left(C+\hat{A}\right)^{-1}.\end{cases}$ (34)

Then $\forall t\in[0,T]$, $\xi(t)$ is invertible and
$a:t\in[0,T]\rightarrow\hat{A}+\xi(t)^{-1}\in\mathcal{S}_{d}(\mathbb{R})$ is
the unique solution of (32).

###### Proof.

First, we easily verify that $\xi$, defined in (33), is solution of the linear
ODE (34). We see that, for all $t\in[0,T]$, $-\xi(t)$ is the sum of
$\frac{\hat{A}^{-1}}{2}\left(I-e^{-2\hat{A}\left(T-t\right)}\right)\in\mathcal{S}^{+}_{d}(\mathbb{R})$
and
$e^{-\hat{A}\left(T-t\right)}\left(C+\hat{A}\right)^{-1}e^{-\hat{A}\left(T-t\right)}\in\mathcal{S}^{++}_{d}(\mathbb{R})$,
so $-\xi(t)\in\mathcal{S}^{++}_{d}(\mathbb{R})$ and $\xi(t)$ is invertible.

We also note that

$a^{\prime}(t)=-\xi(t)^{-1}\xi^{\prime}(t)\xi(t)^{-1}=-\xi(t)^{-1}\hat{A}-\hat{A}\xi(t)^{-1}-\xi(t)^{-2}=\hat{A}^{2}-\left(\hat{A}+\xi(t)^{-1}\right)^{2}=\hat{A}^{2}-a(t)^{2}$

and $a(T)=-C$, hence the result. ∎

We deduce the following corollary:

######  Corollary 1.

$\forall t\in[0,T],\quad
A(t)=\eta^{\frac{1}{2}}\left(\hat{A}-\left(\frac{\hat{A}^{-1}}{2}\left(I-e^{-2\hat{A}(T-t)}\right)+e^{-\hat{A}(T-t)}\left(C+\hat{A}\right)^{-1}e^{-\hat{A}(T-t)}\right)^{-1}\right)\eta^{\frac{1}{2}}.$

##  Appendix B Appendix – Merton portfolio optimization problem under
Ornstein-Uhlenbeck dynamics and exponential utility

###  B.1 Modelling framework

We study in this appendix a Merton problem where prices have multivariate
Ornstein-Uhlenbeck dynamics. It is closely related to our problem and can be
seen as some form of limit case corresponding to no execution costs (i.e.
$L=0$) and no terminal penalty (i.e. $\ell=0$).

The results obtained in this appendix are essential in our proof of existence
of a solution to the system of ODEs (20) on $[0,T]$ with terminal condition
(21) (see Theorem 2).

As in the body of the paper,404040We consider no permanent market impact in
this Appendix. we consider a model with $d$ assets, whose prices are modelled
by a $d$-dimensional stochastic process
$(S_{t})_{t\in[0,T]}=\left(S^{1}_{t},\ldots,S^{d}_{t}\right)_{t\in[0,T]}^{\intercal}$
with dynamics

$dS_{t}=R(\overline{S}-S_{t})dt+VdW_{t},$

where $\overline{S}\in\mathbb{R}^{d}$, $R\in\mathcal{M}_{d}(\mathbb{R})$,
$V\in\mathcal{M}_{d,k}(\mathbb{R})$, and
$(W_{t})_{t\in[0,T]}=\left(W^{1}_{t},\ldots,W^{k}_{t}\right)^{\intercal}_{t\in[0,T]}$
is a $k$-dimensional standard Brownian motion (with independent coordinates),
for some $k\in\mathbb{N}^{*}$. As before, we write $\Sigma=VV^{\intercal}$.

We consider a trader optimizing her portfolio over the period $[0,T]$ by
controlling at each time the number of each asset in her portfolio, i.e. she
controls the $d$-dimensional process
$(q_{t})_{t\in[0,T]}=\left(q^{1}_{t},\ldots,q^{d}_{t}\right)_{t\in[0,T]}^{\intercal}$,
where $q^{i}_{t}$ denotes the number of assets $i$ in the portfolio at time
$t$, for each $i\in\\{1,\ldots,d\\}$ ($t\in[0,T]$).414141Unlike what happens
in the model of Section 2, our control variable is here the number of assets
and not the volume traded. It is only when execution costs are incurred by the
trader that trading rates / trading volumes are indeed the relevant control
variables. The process $(q_{t})_{t\in[0,T]}$ lies in the space of admissible
controls $\mathcal{A}^{Merton}_{0}$, where for $t\in[0,T]$, the set
$\mathcal{A}^{Merton}_{t}$ is defined as

$\displaystyle\mathcal{A}^{Merton}_{t}:=\left\\{(q_{s})_{s\in[t,T]},\
\mathbb{R}^{d}\textrm{-valued},\ \mathbb{F}\textrm{-adapted, satisfying a
linear growth condition with respect to }(S_{s})_{s\in[t,T]}\right\\}.$ (35)

We introduce the process $(\mathcal{V}_{t})_{t\in[0,T]}$ modelling the MtM
value of the trader’s portfolio, i.e.

$\forall
t\in[0,T],\quad\mathcal{V}_{t}=\mathcal{V}_{0}+\int_{0}^{t}q_{s}^{\intercal}dS_{s},\qquad\mathcal{V}_{0}\in\mathbb{R}\
\text{given}.$

For a given $\gamma>0$, the trader aims at maximizing the following objective
function:

$\displaystyle\mathbb{E}\left[-e^{-\gamma\mathcal{V}_{T}}\right],$ (36)

over the set of admissible controls
$(q_{t})_{t\in[0,T]}\in\mathcal{A}^{Merton}_{0}.$ We define her value function
$\hat{u}:[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ as

$\displaystyle\hat{u}(t,\mathcal{V},S)=\underset{(q_{s})_{s\in[t,T]}\in\mathcal{A}^{Merton}_{t}}{\sup}\mathbb{E}\left[-e^{-\gamma\mathcal{V}^{t,\mathcal{V},S,q}_{T}}\right]\qquad\forall(t,\mathcal{V},S)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d},$

where $(\mathcal{V}_{s}^{t,\mathcal{V},S,q})_{s\in[t,T]}$ denotes the process
defined by

$d\mathcal{V}^{t,\mathcal{V},S,q}_{s}=q_{s}^{\intercal}dS^{t,S}_{s},\quad\mathcal{V}^{t,\mathcal{V},S,q}_{t}=\mathcal{V}$

with

$dS^{t,S}_{s}=R(\overline{S}-S^{t,S}_{s})ds+VdW_{s},\quad S^{t,S}_{t}=S.$

###  B.2 HJB equation

The HJB equation associated with Problem (36) is given by

$\displaystyle 0=$
$\displaystyle\,\partial_{t}\hat{w}(t,\mathcal{V},S)+\nabla_{S}\hat{w}(t,\mathcal{V},S)^{\intercal}R(\overline{S}-S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{w}(t,\mathcal{V},S)\right)$ (37)
$\displaystyle+\underset{q\in\mathbb{R}^{d}}{\sup}\left\\{\partial_{\mathcal{V}}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}R(\overline{S}-S)+\frac{1}{2}\partial^{2}_{{\mathcal{V}}{\mathcal{V}}}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}\Sigma
q+\partial_{\mathcal{V}}\nabla_{S}\hat{w}(t,{\mathcal{V}},S)^{\intercal}\Sigma
q\right\\}$

for all $(t,{\mathcal{V}},S)\in[0,T)\times\mathbb{R}\times\mathbb{R}^{d}$,
with terminal condition

$\displaystyle\hat{w}(T,{\mathcal{V}},S)=-e^{-\gamma{\mathcal{V}}}\quad\forall({\mathcal{V}},S)\in\mathbb{R}\times\mathbb{R}^{d}.$
(38)

To solve the above HJB equation, we use the ansatz

$\displaystyle\hat{w}(t,{\mathcal{V}},S)=-e^{-\gamma\left({\mathcal{V}}+\hat{\theta}(t,S)\right)}.$
(39)

Indeed, we have the following proposition:

######  Proposition 4.

If there exists $\hat{\theta}\in
C^{1,2}([0,T]\times\mathbb{R}^{d},\mathbb{R})$ solution to

$\displaystyle 0=\ $
$\displaystyle\partial_{t}\hat{\theta}(t,S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{\theta}(t,S)\right)+\frac{1}{2\gamma}(\overline{S}-S)^{\intercal}R^{\intercal}\Sigma^{-1}R(\overline{S}-S)$
(40)

on $[0,T)\times\mathbb{R}^{d}$, with terminal condition

$\displaystyle\hat{\theta}(T,S)=0\qquad\forall S\in\mathbb{R}^{d},$ (41)

then the function
$\hat{w}:[0,T]\times\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}$
defined by

$\hat{w}(t,{\mathcal{V}},S)=-e^{-\gamma\left({\mathcal{V}}+\hat{\theta}(t,S)\right)}\quad\forall(t,{\mathcal{V}},S)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$

is a solution to (37) on $[0,T)\times\mathbb{R}\times\mathbb{R}^{d}$ with
terminal condition (38).

###### Proof.

Let $\hat{\theta}:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a
solution to (40) on $[0,T)\times\mathbb{R}^{d}$ with terminal condition (41),
then we have for all
$(t,{\mathcal{V}},S)\in[0,T)\times\mathbb{R}\times\mathbb{R}^{d}$:

$\displaystyle\partial_{t}\hat{w}(t,{\mathcal{V}},S)+\nabla_{S}\hat{w}(t,{\mathcal{V}},S)^{\intercal}R(\overline{S}-S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{w}(t,{\mathcal{V}},S)\right)$
$\displaystyle+\underset{q\in\mathbb{R}^{d}}{\sup}\left\\{\partial_{\mathcal{V}}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}R(\overline{S}-S)+\frac{1}{2}\partial^{2}_{{\mathcal{V}}{\mathcal{V}}}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}\Sigma
q+\partial_{\mathcal{V}}\nabla_{S}\hat{w}(t,{\mathcal{V}},S)^{\intercal}\Sigma
q\right\\}$ $\displaystyle=\ $
$\displaystyle-\gamma\partial_{t}\hat{\theta}(t,S)\hat{w}(t,{\mathcal{V}},S)-\gamma\nabla_{S}\hat{\theta}(t,S)R(\overline{S}-S)\hat{w}(t,{\mathcal{V}},S)-\gamma\hat{w}(t,{\mathcal{V}},S)\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{\theta}(t,S)\right)$
$\displaystyle+\frac{\gamma^{2}}{2}\hat{w}(t,{\mathcal{V}},S)\nabla_{S}\hat{\theta}(t,S)^{\intercal}\Sigma\nabla_{S}\hat{\theta}(t,S)$
$\displaystyle+\underset{q\in\mathbb{R}^{d}}{\sup}\left\\{-\gamma\hat{w}(t,{\mathcal{V}},S)q^{\intercal}R(\overline{S}-S)+\frac{\gamma^{2}}{2}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}\Sigma
q+\gamma^{2}\hat{w}(t,{\mathcal{V}},S)\nabla_{S}\hat{\theta}(t,{\mathcal{V}},S)^{\intercal}\Sigma
q\right\\}$ $\displaystyle=\ $
$\displaystyle-\gamma\hat{w}(t,{\mathcal{V}},S)\Bigg{(}\partial_{t}\hat{\theta}(t,S)+\nabla_{S}\hat{\theta}(t,S)R(\overline{S}-S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{\theta}(t,S)\right)-\frac{\gamma}{2}\nabla_{S}\hat{\theta}(t,S)^{\intercal}\Sigma\nabla_{S}\hat{\theta}(t,S)$
$\displaystyle+\underset{q\in\mathbb{R}^{d}}{\sup}\left\\{q^{\intercal}\left(R(\overline{S}-S)-\gamma\Sigma\nabla_{S}\hat{\theta}(t,S)\right)-\frac{\gamma}{2}q^{\intercal}\Sigma
q\right\\}\Bigg{)}.$

The supremum in the above equation is reached for
$q=q^{*}(t,S)=\frac{1}{\gamma}\Sigma^{-1}R(\overline{S}-S)-\nabla_{S}\hat{\theta}(t,S)$,
and we obtain therefore after simplifications:

$\displaystyle\partial_{t}\hat{w}(t,{\mathcal{V}},S)+\nabla_{S}\hat{w}(t,{\mathcal{V}},S)^{\intercal}R(\overline{S}-S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{w}(t,{\mathcal{V}},S)\right)$
$\displaystyle+\underset{q\in\mathbb{R}^{d}}{\sup}\left\\{\partial_{\mathcal{V}}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}R(\overline{S}-S)+\frac{1}{2}\partial^{2}_{{\mathcal{V}}{\mathcal{V}}}\hat{w}(t,{\mathcal{V}},S)q^{\intercal}\Sigma
q+\partial_{\mathcal{V}}\nabla_{S}\hat{w}(t,{\mathcal{V}},S)^{\intercal}\Sigma
q\right\\}$ $\displaystyle=\ $
$\displaystyle-\gamma\hat{w}(t,{\mathcal{V}},S)\Bigg{(}\partial_{t}\hat{\theta}(t,S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{\theta}(t,S)\right)+\frac{1}{2\gamma}(\overline{S}-S)^{\intercal}R^{\intercal}\Sigma^{-1}R(\overline{S}-S)\Bigg{)}$
$\displaystyle=\ $ $\displaystyle 0.$

As $\hat{w}$ satisfies the terminal condition (38), the result is proved. ∎

We now use a second ansatz and look for a function $\hat{\theta}$ solution to
(40) on $[0,T)\times\mathbb{R}^{d}$ with terminal condition (41) of the
following form:

$\displaystyle\hat{\theta}(t,S)=S^{\intercal}\hat{C}(t)S+\hat{E}(t)^{\intercal}S+\hat{F}(t).$
(42)

We have indeed the following proposition:

######  Proposition 5.

Assume there exists $\hat{C}\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $\hat{E}\in
C^{1}\left([0,T],\mathbb{R}^{d}\right)$, $\hat{F}\in
C^{1}\left([0,T],\mathbb{R}\right)$ satisfying the system of ODEs

$\displaystyle\begin{cases}\hat{C}^{\prime}(t)&=-\frac{1}{2\gamma}R^{\intercal}\Sigma^{-1}R\\\
\hat{E}^{\prime}(t)&=\frac{1}{\gamma}R^{\intercal}\Sigma^{-1}R\overline{S}\\\
\hat{F}^{\prime}(t)&=-\text{Tr}\left(\hat{C}(t)\Sigma\right)-\frac{1}{2\gamma}\overline{S}^{\intercal}R^{\intercal}\Sigma^{-1}R\overline{S},\end{cases}$
(43)

with terminal condition

$\displaystyle\hat{C}(T)=\hat{E}(T)=\hat{F}(T)=0.$ (44)

Then the function $\hat{\theta}$ defined by (42) satisfies (40) on
$[0,T)\times\mathbb{R}^{d}$ with terminal condition (41).

###### Proof.

Let us consider $\hat{C}\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $\hat{E}\in
C^{1}\left([0,T],\mathbb{R}^{d}\right)$, $\hat{F}\in
C^{1}\left([0,T],\mathbb{R}\right)$ verifying (43) on $[0,T)$ with terminal
condition (44). Let us consider
$\hat{\theta}:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ defined by (42).
Then we obtain for all $(t,S)\in[0,T)\times\mathbb{R}^{d}$:

$\displaystyle\partial_{t}\hat{\theta}(t,S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{\theta}(t,S)\right)+\frac{1}{2\gamma}(\overline{S}-S)^{\intercal}R^{\intercal}\Sigma^{-1}R(\overline{S}-S)$
$\displaystyle=\quad$ $\displaystyle
S^{\intercal}\hat{C}^{\prime}(t)S+{\hat{E}^{\prime}(t)}^{\intercal}S+\hat{F}^{\prime}(t)+\text{Tr}\left(\hat{C}(t)\Sigma\right)+\frac{1}{2\gamma}(\overline{S}-S)^{\intercal}R^{\intercal}\Sigma^{-1}R(\overline{S}-S)$
$\displaystyle=\quad$ $\displaystyle 0.$

As it is straightforward to verify that $\hat{\theta}$ satisfies the terminal
condition (41), the result is proved. ∎

It is straightforward to see that there exists a unique solution $\hat{C}\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $\hat{E}\in
C^{1}\left([0,T],\mathbb{R}^{d}\right)$, $\hat{F}\in
C^{1}\left([0,T],\mathbb{R}\right)$ to (43) with terminal condition (44). We
can then prove the following verification theorem.

######  Theorem 3.

We consider the functions $\hat{C}\in
C^{1}\left([0,T],\mathcal{S}_{d}(\mathbb{R})\right)$, $\hat{E}\in
C^{1}\left([0,T],\mathbb{R}^{d}\right)$, $\hat{F}\in
C^{1}\left([0,T],\mathbb{R}\right)$ solutions to (43) with terminal condition

$\hat{C}(T)=\hat{E}(T)=\hat{F}(T)=0,$

i.e. for all $t\in[0,T]$,

$\begin{cases}\hat{C}(t)=\frac{1}{2\gamma}(T-t)R^{\intercal}\Sigma^{-1}R,\\\
\hat{E}(t)=-\frac{1}{\gamma}(T-t)R^{\intercal}\Sigma^{-1}R\overline{S},\\\
\hat{F}(t)=\frac{1}{4\gamma}(T-t)^{2}\text{Tr}\left(R^{\intercal}\Sigma^{-1}R\Sigma\right)+\frac{1}{2\gamma}(T-t)\overline{S}^{\intercal}R^{\intercal}\Sigma^{-1}R.\end{cases}$

We consider the function $\hat{\theta}$ defined by

$\hat{\theta}(t,S)=S^{\intercal}\hat{C}(t)S+\hat{E}(t)^{\intercal}S+\hat{F}(t),$

and the associated function $\hat{w}$ defined by

$\hat{w}(t,\mathcal{V},S)=-e^{-\gamma\left(\mathcal{V}+\hat{\theta}(t,S)\right)}.$

For all $(t,\mathcal{V},S)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$ and
$q=(q_{s})_{s\in[t,T]}\in\mathcal{A}^{Merton}_{t}$, we have

$\displaystyle\mathbb{E}\left[-e^{-\gamma\mathcal{V}^{t,\mathcal{V},S,q}_{T}}\right]\leq\hat{w}(t,\mathcal{V},S).$
(45)

Moreover, equality is obtained in (45) by taking the optimal control
$(q^{*}_{s})_{s\in[t,T]}\in\mathcal{A}^{Merton}_{t}$ given by the closed-loop
feedback formula

$\displaystyle
q^{*}_{s}=\frac{1}{\gamma}(I_{d}+(T-s)R^{\intercal})\Sigma^{-1}R(\overline{S}-S_{s}^{t,S}).$
(46)

In particular, $\hat{w}=\hat{u}.$

###### Proof.

It is obvious that $(q^{*}_{s})_{s\in[t,T]}\in\mathcal{A}^{Merton}_{t}$ (i.e.,
$(q^{*}_{s})_{s\in[t,T]}$ is well defined and admissible):

$\displaystyle\exists C_{t,T}>0,\forall s\in[t,T],\hskip 28.45274pt\parallel
q^{*}_{s}\parallel\leqslant
C_{t,T}\left(1+\underset{\tau\in[t,s]}{\sup}\parallel
S_{\tau}\parallel\right).$

Let us consider
$(t,\mathcal{V},S)\in[0,T]\times\mathbb{R}\times\mathbb{R}^{d}$ and
$q=(q_{s})_{s\in[t,T]}\in\mathcal{A}^{Merton}_{t}$. We now prove that

$\displaystyle\mathbb{E}\left[\hat{w}\left(T,\mathcal{V}^{t,\mathcal{V},S}_{T},S^{t,S}_{T}\right)\right]\leqslant\hat{w}(t,\mathcal{V},S).$

We use the following notations for readability

$\displaystyle\forall s\in[t,T],\hskip
14.22636pt\hat{w}\left(s,\mathcal{V}^{t,\mathcal{V},S,q}_{s},S^{t,S}_{s}\right)=\hat{w}^{t,\mathcal{V},S,q}_{s},$
$\displaystyle\forall s\in[t,T],\hskip
14.22636pt\hat{\theta}(s,S^{t,S}_{s})=\hat{\theta}^{t,S}_{s}.$

By Itô’s formula, we have $\forall s\in[0,T]$

$\displaystyle
d\hat{w}^{t,\mathcal{V},S,q}_{s}=\mathcal{L}^{q}\hat{w}^{t,\mathcal{V},S,q}_{s}ds+\left(\partial_{\mathcal{V}}\hat{w}^{t,\mathcal{V},S,q}_{s}q_{s}+\nabla_{S}\hat{w}^{t,\mathcal{V},S,q}_{s}\right)^{\intercal}VdW_{s},$

where

$\displaystyle\mathcal{L}^{q}\hat{w}^{t,\mathcal{V},S,q}_{s}$ $\displaystyle=$
$\displaystyle\partial_{t}w^{t,\mathcal{V},S,q}_{s}+\left(\nabla_{S}{\hat{w}}^{t,\mathcal{V},S,q}_{s}\right)^{\intercal}R(\overline{S}-S)+\partial_{\mathcal{V}}\hat{w}^{t,\mathcal{V},S,q}_{s}q^{\intercal}_{s}R(\overline{S}-S)+\frac{1}{2}\text{Tr}\left(\Sigma
D^{2}_{SS}\hat{w}^{t,\mathcal{V},S,q}_{s}\right)$
$\displaystyle+\frac{1}{2}\partial^{2}_{\mathcal{V}\mathcal{V}}\hat{w}^{t,\mathcal{V},S,q}_{s}q_{s}^{\intercal}\Sigma
q_{s}+\left(\partial_{\mathcal{V}}\nabla_{S}{\hat{w}}^{t,\mathcal{V},S,q}_{s}\right)^{\intercal}\Sigma
q_{s}.$

We have

$\displaystyle\nabla_{S}\hat{w}^{t,\mathcal{V},S,q}_{s}$
$\displaystyle=-\gamma\hat{w}^{t,\mathcal{V},S,q}_{s}\nabla_{S}\theta^{t,S}_{s}$
$\displaystyle=-\gamma\hat{w}^{t,\mathcal{V},S,q}_{s}\left(2\hat{C}(s)S^{t,S}_{s}+\hat{E}(s)\right),$

and

$\displaystyle\partial_{\mathcal{V}}\hat{w}^{t,\mathcal{V},S,q}_{s}$
$\displaystyle=-\gamma\hat{w}^{t,\mathcal{V},S,q}_{s}.$

We define for all $s\in[t,T]$

$\displaystyle\kappa^{q}_{s}$
$\displaystyle=-\gamma\left(q_{s}+2\hat{C}(s)S^{t,S}_{s}+\hat{E}(s)\right),$
$\displaystyle\xi^{q}_{t,s}$
$\displaystyle=\exp\left(\int_{t}^{s}{\kappa^{q}_{\varrho}}^{\intercal}VdW_{\varrho}-\frac{1}{2}\int_{t}^{s}{\kappa^{q}_{\varrho}}^{\intercal}\Sigma\kappa^{q}_{\varrho}d\varrho\right).$

We then have

$\displaystyle
d\left(\hat{w}^{t,\mathcal{V},S,q}_{s}\left(\xi^{q}_{t,s}\right)^{-1}\right)=\left(\xi^{q}_{t,s}\right)^{-1}\mathcal{L}^{q}\hat{w}^{t,\mathcal{V},S,q}_{s}ds.$

By definition of $\hat{w}$,
$\mathcal{L}^{q}\hat{w}^{t,\mathcal{V},S,q}_{s}\leqslant 0$.

Moreover, equality holds for the control reaching the supremum in (37). It is
easy to see that the supremum is reached for the unique value

$\displaystyle q_{s}$
$\displaystyle=\frac{1}{\gamma}\Sigma^{-1}R(\overline{S}-S^{t,S}_{s})-\nabla_{S}\hat{\theta}(t,S^{t,S}_{s})$
$\displaystyle=\frac{1}{\gamma}\Sigma^{-1}R(\overline{S}-S^{t,S}_{s})-2\hat{C}(s)S^{t,S}_{s}-\hat{E}(s)$
$\displaystyle=\frac{1}{\gamma}(I_{d}+(T-s)R^{\intercal})\Sigma^{-1}R(\overline{S}-S_{s}^{t,S})$

which corresponds to $(q_{s})_{s\in[t,T]}=(q^{*}_{s})_{s\in[t,T]}$.

As a consequence,
$\left(\hat{w}^{t,\mathcal{V},S,q}_{s}\left(\xi^{q}_{t,s}\right)^{-1}\right)_{s\in[t,T]}$
is nonincreasing and therefore

$\displaystyle\hat{w}\left(T,\mathcal{V}^{t,\mathcal{V},S,q}_{T},S^{t,S}_{T}\right)\leqslant\hat{w}(t,\mathcal{V},S)\xi^{q}_{t,T},$

with equality when $(q_{s})_{s\in[t,T]}=(q^{*}_{s})_{s\in[t,T]}$.

Taking expectation we get

$\displaystyle\mathbb{E}\left[\hat{w}\left(T,\mathcal{V}^{t,\mathcal{V},S,q}_{T},S^{t,S}_{T}\right)\right]\leqslant\hat{w}(t,\mathcal{V},S)\mathbb{E}\left[\xi^{q}_{t,T}\right].$

We proceed to prove that $\mathbb{E}\left[\xi^{q}_{t,T}\right]$ is equal to 1.
To do so, we use that $\xi^{q}_{t,t}=1$ and prove that
$(\xi^{q}_{t,s})_{s\in[t,T]}$ is a martingale under
$\left(\mathbb{P};\mathbb{F}=(\mathcal{F}_{s})_{s\in[t,T]}\right)$.

We know that $(q^{t,q}_{s})_{s\in[t,T]}$ satisfies a linear growth condition
with respect to $(S^{t,S}_{s})_{s\in[t,T]}$. Given the form of $\kappa$ one
can easily show that there exists a constant $C$ such that

$\displaystyle\underset{s\in[t,T]}{\sup}{\parallel\kappa^{q}_{s}\parallel}^{2}\leqslant
C\left(1+\underset{s\in[t,T]}{\sup}{\parallel
W_{s}-W_{t}\parallel}^{2}\right).$

By using classical properties of the Brownian motion, we prove that

$\displaystyle\exists\epsilon>0,\forall s\in[t,T],\hskip
8.5359pt\mathbb{E}\left[\exp\left(\frac{1}{2}\int_{s}^{\left(s+\epsilon\right)\wedge
T}{\kappa^{q}_{\varrho}}^{\intercal}\Sigma\kappa^{q}_{\varrho}d\varrho\right)\right]<+\infty.$

Using a classical trick due to Beneš (see [29], Chapter 5), we see that
$(\xi^{q}_{t,s})_{s\in[t,T]}$ is a martingale under
$\left(\mathbb{P};\mathbb{F}=(\mathcal{F}_{s})_{s\in[t,T]}\right)$.

We obtain

$\displaystyle\mathbb{E}\left[\hat{w}\left(T,\mathcal{V}^{t,\mathcal{V},S,q}_{T},S^{t,S}_{T}\right)\right]\leqslant\hat{w}(t,\mathcal{V},S),$

with equality when $(q_{s})_{s\in[t,T]}=(q^{*}_{s})_{s\in[t,T]}$.

We conclude that

$\displaystyle\hat{u}\left(t,\mathcal{V},S\right)$
$\displaystyle=\underset{(q_{s})_{s\in[t,T]}\in\mathcal{A}^{Merton}_{t}}{\sup}\mathbb{E}\left[-\exp\left(-\gamma
V_{T}^{t,\mathcal{V},S,q}\right)\right]$
$\displaystyle=\mathbb{E}\left[-\exp\left(-\gamma
V_{T}^{t,\mathcal{V},S,q^{*}}\right)\right]$
$\displaystyle=\hat{w}(t,\mathcal{V},S).$

∎

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