# Optimal trade execution under small market impact and portfolio liquidation
with semimartingale strategies††thanks: Financial support through the Stiftung
der Deutschen Wirtschaft (sdw) is gratefully acknowledged. We thank Peter
Bank, Mikhail Urusov and seminar participants at various institutions for
valuable comments and remarks. This is a substantially extended version of a
previous preprint entitled Small impact analysis in stochastically illiquid
markets.

Ulrich Horst111Department of Mathematics, and School of Business and
Economics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin,
Germany<EMAIL_ADDRESS>Evgueni Kivman222Department of Mathematics,
Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany,
<EMAIL_ADDRESS>
###### Abstract

We consider an optimal liquidation problem with instantaneous price impact and
stochastic resilience for small instantaneous impact factors. Within our
modelling framework, the optimal portfolio process converges to the solution
of an optimal liquidation problem with general semimartingale controls when
the instantaneous impact factor converges to zero. Our results provide a
unified framework within which to embed the two most commonly used modelling
frameworks in the liquidation literature and provide a microscopic foundation
for the use of semimartingale liquidation strategies and the use of portfolio
processes of unbounded variation. Our convergence results are based on novel
convergence results for BSDEs with singular terminal conditions and novel
representation results of BSDEs in terms of uniformly continuous functions of
forward processes.

AMS Subject Classification: 93E20, 91B70, 60H30.

Keywords: portfolio liquidation, singular BSDE, stochastic liquidity, singular
control

##  1 Introduction

The impact of limited liquidity on optimal trade execution has been
extensively analysed in the mathematical finance and stochastic control
literature in recent years. The majority of the optimal portfolio liquidation
literature allows for one of two possible price impacts. The first approach,
pioneered by Bertsimas and Lo [6] and Almgren and Chriss [3], divides the
price impact in a purely temporary effect, which depends only on the present
trading rate and does not influence future prices, and in a permanent effect,
which influences the price depending only on the total volume that has been
traded in the past. The temporary impact is typically assumed to be linear in
the trading rate, leading to a quadratic term in the cost functional. The
original modelling framework has been extended in various directions including
general stochastic settings with and without model uncertainty and multi-
player and mean-field-type models by many authors including Ankirchner et al.
[4], Cartea and Jaimungal [10], Fu et al. [14], Gatheral and Schied [16],
Graewe et al. [18], Horst et al. [22], Kruse and Popier [24] and Neuman and
Voß [29].

A second approach, initiated by Obizhaeva and Wang [30], assumes that price
impact is not permanent, but transient with the impact of past trades on
current prices decaying over time. When impact is transient, one often allows
for both absolutely continuous and singular trading strategies. When singular
controls are admissible, optimal liquidation strategies usually comprise large
block trades at the initial and terminal time. The work of Obizhaeva and Wang
has been extended by Alfonsi et al. [2], Chen et al. [11], Fruth et al. [13],
Gatheral [15], Guéant [19], Horst and Naujokat [20], Lokka and Xu [27] and
Predoiu et al. [31], among many others.

Single and multi-asset liquidation problems with instantaneous and transient
market impact and stochastic resilience333Empirical work of Large [25] and Lo
and Hall [26] suggests that resilience of market impact does indeed vary
stochastically. where trading is confined to absolutely continuous strategies
have been analyzed in Graewe and Horst [17] and Horst and Xia [21],
respectively. Although only absolutely continuous trading strategies were
admissible in both papers, numerical simulations reported in [17] suggest that
if all model parameters are deterministic constants, as the instantaneous
impact parameter converges to zero444Cartea and Jaimungal [9] provide
empirical evidence that the instantaneous price impact is indeed (much)
smaller than permanent (or transient) price impact., the optimal portfolio
process converges to the optimal solution in [30] with two block trades and a
constant trading rate. This suggests that the model in [17] interpolates
between the two most commonly used liquidation models [3, 30].

This paper provides a rigorous convergence analysis for small instantaneous
impact within a Markovian factor model. It turns out that the stochastic
setting is quite different from the deterministic one. Most importantly, we
show that in the stochastic setting, the optimal portfolio processes obtained
in [17] converge (in the sense of graph convergence of the completed graphs in
the Hausdorff distance) to a process of infinite variation with jumps as the
instantaneous market impact parameter converges to zero. Subsequently, we
prove that the limiting portfolio process is optimal in a liquidation model
with general semimartingale execution strategies.

Within a portfolio liquidation framework, inventory processes with infinite
variation were first considered by Lorenz and Schied [28] to the best of our
knowledge. Later, Becherer et al. [5] considered a trading framework with
generalized price impact and proved that the cost functional depends
continuously on the trading strategy, considered in several topologies.
Bouchard et al. [7] considered infinite variation inventory processes in the
context of hedging. More recently, Ackermann et al. [1] considered a
liquidation model with general RCLL semimartingale trading strategies. This
paper provides a microscopic foundation for the use of semimartingale
strategies.

Our work also complements empirical work of Carmona and Webster [8], who
provide strong evidence that inventories of large traders often do have indeed
a non-trivial quadratic variation component. For instance, for the Blackberry
stock, they analyze the inventories of “the three most active Blackberry
traders” on a particular day, namely CIBC World Markets Inc., Royal Bank of
Canada Capital Markets, and TD Securities Inc. From their data, they “suspect
that RBC (resp. TD Securities) were trading to acquire a long (resp. short)
position in Blackberry” and found that the corresponding inventory processes
were with infinite variation. More generally, they find that systematic tests
“on different days and other actively traded stocks give systematic rejections
of this null hypothesis [quadratic variation of inventory being zero], with a
$p$-value never greater than $10^{-5}$.” Our results suggest that inventories
with non-trivial quadratic variation arise naturally when market depth is high
and resilience and/or market risk fluctuates stochastically. This is very
intuitive; in deep markets it is comparably cheap to frequently adjust
portfolios to stochastically varying market environments.

The main technical challenge in establishing our convergence results is that
the optimal solution to the limiting model cannot be obtained by taking the
limit of the three-dimensional quadratic BSDE system that characterizes the
optimal solution in the model with positive instantaneous impact. Instead, we
prove that the limit is fully characterized by the solution to a one-
dimensional quadratic BSDE. Remarkably, this BSDE is independent of the
liquidation requirement. As a result, full liquidation takes place if the
instantaneous impact parameter converges to zero even it is not strictly
required. The reason is a loss in book value of the remaining shares that
outweighs the liquidation cost for small instantaneous impact. Our convergence
result is based on a novel representation result for solutions for BSDEs
driven by Itô processes in terms of uniformly continuous functions of the
forward process and on a series of novel convergence results for sequences of
singular stochastic integral equations and random ODEs, which we choose to
report in an abstract setting in Appendix A and B, respectively.

The limiting portfolio process is optimal in a liquidation model with general
semimartingale execution strategies. Within our of modeling framework where
the cost coefficients are driven by continuous factor processes block trades
optimally occur only at the beginning and the end of the trading period. This
is very intuitive as one would expect large block trades to require some form
trigger such as an external shock leading to a discontinuous change of cost
coefficients. The proof of optimality proceeds in three steps. We first prove
that the process with jumps can be approximated by absolutely continuous ones.
This allows us to approximate the trading costs in the semimartingale by
trading costs in the pre-limit models from which we finally deduce the
optimality of the limiting process in the semimartingale model by using the
optimality of the approximating inventory processes in the pre-limit models.

The rest of the paper is organized as follows. In Section 2, we recall the
modelling setup from [17, 21] and summarize our main results. The proofs are
given in Sections 3 and 4. A series of abstract convergence results for
various stochastic equation with singularities upon which our convergence
results are based is postponed to two appendices.

Notation. Throughout, randomness is described by an $\mathbb{R}^{m}$-valued
Brownian motion $\\{W_{t}\\}_{t\in[0;T]}$ defined on a complete probability
space $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\in[0;T]},\mathbb{P})$
where $\\{\mathcal{F}_{t}\\}_{t\in[0;T]}$ denotes the filtration generated by
$W$, augmented by the $\mathbb{P}$-null-sets. Unless otherwise specified, all
equations and inequalities hold in the $\mathbb{P}$-a.s. sense. For a subset
$A\subseteq\mathbb{R}^{d}$, we denote by
$\mathcal{L}_{\textup{Prog}}^{2}(\Omega\times[0;T];A)$ the set of all
progressively measurable $A$-valued stochastic processes
$\\{X_{t}\\}_{t\in[0;T]}$ such that
$\mathbb{E}(\int^{T}_{0}\|X_{t}\|_{2}^{2}\mathrm{d}t)<\infty,$ while
$\mathcal{L}_{\textup{Prog}}^{\infty}(\Omega\times[0;T];A)$ denotes the subset
of essentially bounded processes.
$\mathcal{L}_{\mathcal{P}}^{2}(\Omega\times[0;T];A)$ and
$\mathcal{L}_{\mathcal{P}}^{\infty}(\Omega\times[0;T];A)$ denote the
respective subsets of predictable processes. Whenever $T-$ appears, we mean
that there exists an $\varepsilon>0$ such that a statement holds for all
$T^{\prime}\in(T-\varepsilon;T)$.

##  2 Problem formulation and main results

In this section, we introduce two portfolio liquidation models with stochastic
market impact. In the first model, analyzed in Section 2.1, the investor is
confined to absolutely continuous trading strategies. For small instantaneous
market impact, we prove that the optimal liquidation strategy converges to a
semimartingale with jumps. In Section 2.2, we therefore analyze a liquidation
model with semimartingale trading strategies. We prove that the limiting
process obtained in Section 2.1 is optimal in a model where semimartingale
strategies that satisfy a suitable regularity condition are admissible.

###  2.1 Portfolio liquidation with absolutely continuous strategies

We take the liquidation model analyzed in [17, 21] as our starting point and
consider an investor that needs to close a (single-asset) portfolio of
$x_{0}>0$ shares within a given time interval $[0;T]$ using a trading strategy
$\xi=\\{\xi_{t}\\}_{t\in[0;T]}$. If $\xi_{t}<0$, the investor is selling the
asset at a rate $\xi_{t}$ at time $t\in[0;T]$, else she is buying it. For a
given strategy $\xi$, the corresponding portfolio process
$X^{\xi}=\\{X^{\xi}_{t}\\}_{t\in[0;T]}$ satisfies the ODE

$\displaystyle\mathrm{d}X^{\xi}_{t}=\xi_{t}\mathrm{d}t,\quad
X^{\xi}_{0}=x_{0}.$

The set of admissible strategies is given by the set

$\mathcal{A}:=\mathcal{L}_{\textup{Prog}}^{2}(\Omega\times[0;T];\mathbb{R}).$

The transient price impact corresponding to a general inventory process
$X\in\mathcal{L}_{\mathcal{P}}^{2}(\Omega\times[0;T];\mathbb{R})$ is described
by the unique stochastic process $Y^{X}=\\{Y^{X}_{t}\\}_{t\in[0;T]}$, which
satisfies the ODE

$\displaystyle\mathrm{d}Y^{X}_{t}=\gamma\mathrm{d}X_{t}-\rho_{t}Y^{X}_{t}\mathrm{d}t,\quad
Y^{X}_{0}=0$ (2.1)

for some constant $\gamma>0$ and some essentially bounded, adapted,
$(0;\infty)$-valued process $\rho=\\{\rho_{t}\\}_{t\in[0;T]}$. The process
$Y^{X}$ may be viewed as describing an additional shift in the unaffected
benchmark price process generated by the large investor’s trading activity.
For $\xi\in\mathcal{A}$, we write $Y^{\xi}:=Y^{X^{\xi}}$.

For any penalization factor $N\in\mathcal{N}:=[\underline{N};\infty]$555The
case $N=\infty$ captures the case where full liquidation is required; this
case is analyzed in [17]. The case $\gamma+1\leq N<\infty$ is analyzed in
[21]., where

$\underline{N}:=\gamma+1+\sqrt{2\eta\max\big{(}\|\lambda\|_{L^{\infty}},\gamma\|\rho\|_{L^{\infty}}\big{)}},$

and any instantaneous impact factor $\eta>0$, the cost functional is given by

$\displaystyle
J^{\eta,N}(\xi)=\mathbb{E}\bigg{(}\frac{\eta}{2}\int^{T}_{0}(\xi_{t})^{2}\mathrm{d}t+\int^{T}_{0}Y^{\xi}_{t}\mathrm{d}X^{\xi}_{t}+\frac{1}{2}\int^{T}_{0}\lambda_{t}\big{(}X^{\xi}_{t}\big{)}^{2}\mathrm{d}t+\frac{N}{2}\big{(}X^{\xi}_{T}\big{)}^{2}-X^{\xi}_{T}Y^{\xi}_{T}\bigg{)}.$

The first term in the above cost functional captures the instantaneous trading
costs; the second captures the costs from transient price impact; the third
captures market risk where the adapted and non-negative process
$\lambda=\\{\lambda_{t}\\}_{t\in[0;T]}$ specifies the degree of risk. If full
liquidation is required ($N=\infty$), the fourth term should formally be read
as $+\infty\mathbbm{1}_{\\{X^{\xi}_{T}\neq 0\\}}$ with the convention that
$0\cdot\infty=0$. The fifth term captures an additional loss in book value of
the remaining shares. It drops out of the cost function if $N=\infty$; see
[17, 21] for further details on the impact costs and cost coefficients.

It has been shown in [17, 21] that the optimization problem

$\displaystyle\min_{\xi\in\mathcal{A}}J^{\eta,N}(\xi)$ (2.2)

has a solution $\hat{\xi}^{\eta,N}$ for any $N\in{\cal N}$ and any $\eta>0$.
The solution is given in terms of a backward SDE system with possibly singular
terminal condition. We index the optimal trading strategies and state
processes by $\eta$ and $N$ as we are interested in their behavior for small
instantaneous impact factors for both finite and infinite $N$.

######  Theorem 2.1 ([17, 21]).

With the above assumptions, for all $\eta\in(0;\infty)$ and $N\in\mathcal{N}$,
the following holds.

  * i)

The BSDE system

$\begin{split}-dA^{\eta,N}_{t}=&\bigg{(}\lambda_{t}-\frac{1}{\eta}\big{(}A^{\eta,N}_{t}-\gamma
B^{\eta,N}_{t}\big{)}^{2}\bigg{)}\mathrm{d}t-Z^{\eta,N,A}_{t}\mathrm{d}W_{t},\\\
-dB^{\eta,N}_{t}=&\bigg{(}-\rho_{t}B^{\eta,N}_{t}+\frac{1}{\eta}\big{(}\gamma
C^{\eta,N}_{t}-B^{\eta,N}_{t}+1\big{)}\big{(}A^{\eta,N}_{t}-\gamma
B^{\eta,N}_{t}\big{)}\bigg{)}\mathrm{d}t-Z^{\eta,N,B}_{t}\mathrm{d}W_{t},\\\
-dC^{\eta,N}_{t}=&\bigg{(}-2\rho_{t}C^{\eta,N}_{t}-\frac{1}{\eta}\big{(}\gamma
C^{\eta,N}_{t}-B^{\eta,N}_{t}+1\big{)}^{2}\bigg{)}\mathrm{d}t-Z^{\eta,N,C}_{t}\mathrm{d}W_{t}\end{split}$
(2.3)

with terminal condition

$\displaystyle B^{\eta,N}_{T}=1,\quad C^{\eta,N}_{T}=0$ (2.4)

together with $A^{\eta,N}_{T}=N$ in the case $N<\infty$ and $\lim_{t\nearrow
T}A^{\eta,\infty}_{t}=\infty$ in $L^{\infty}$ if $N=\infty$ has a solution

$\left((A^{\eta,N},B^{\eta,N},C^{\eta,N}),(Z^{\eta,N,A},Z^{\eta,N,B},Z^{\eta,N,C})\right)$

that belongs to the space
$\mathcal{L}^{\infty}_{\mathcal{P}}(\Omega\times[0;T];\mathbb{R}^{3})\times\mathcal{L}^{2}_{\textup{Prog}}(\Omega\times[0;T];\mathbb{R}^{3\times
m})$ if $N<\infty$ and to the space
$\mathcal{L}^{\infty}_{\mathcal{P}}(\Omega\times[0;T-];\mathbb{R}^{3})\times\mathcal{L}^{2}_{\textup{Prog}}(\Omega\times[0;T-];\mathbb{R}^{3\times
m})$ if $N=\infty$.

  * ii)

The liquidation problem (2.2) has a solution $\hat{\xi}^{\eta,N}$. The
corresponding state process

$(\hat{X}^{\eta,N},\hat{Y}^{\eta,N}):=(X^{\hat{\xi}^{\eta,N}},Y^{\hat{\xi}^{\eta,N}})$

is given by the (unique) solution to the ODE system

$\begin{split}\partial_{t}\hat{X}^{\eta,N}_{t}=&-\frac{1}{\sqrt{\eta}}D^{\eta,N}_{t}\hat{X}^{\eta,N}_{t}-\frac{1}{\sqrt{\eta}}E^{\eta,N}_{t}\hat{Y}^{\eta,N}_{t},\\\
\partial_{t}\hat{Y}^{\eta,N}_{t}=&-\frac{\gamma}{\sqrt{\eta}}D^{\eta,N}_{t}\hat{X}^{\eta,N}_{t}-\frac{\gamma}{\sqrt{\eta}}E^{\eta,N}_{t}\hat{Y}^{\eta,N}_{t}-\rho_{t}\hat{Y}^{\eta,N}_{t}\end{split}$
(2.5)

with initial conditions $\hat{X}^{\eta,N}_{0}=x_{0}$ and
$\hat{Y}^{\eta,N}_{0}=0$ where

$\displaystyle D^{\eta,N}:=\frac{1}{\sqrt{\eta}}\big{(}A^{\eta,N}-\gamma
B^{\eta,N}\big{)},\quad E^{\eta,N}:=\frac{1}{\sqrt{\eta}}\big{(}\gamma
C^{\eta,N}-B^{\eta,N}+1\big{)}.$

We now state an additional result on the optimal state process, which will be
important for our subsequent analysis. In particular, we show that the optimal
portfolio process $\hat{X}^{\eta,N}$ never changes its sign. The proof is
given in Section 3.1.1.

######  Theorem 2.2.

For all $\eta\in(0;\infty)$, $N\in\mathcal{N}$, the process

$\hat{Z}^{\eta,N}:=\gamma\hat{X}^{\eta,N}-\hat{Y}^{\eta,N}$

is non-increasing on $[0;T]$. Moreover,

$\displaystyle\mathbb{P}\Big{[}\hat{X}^{\eta,N}_{t}\in(0;x_{0}),\leavevmode\nobreak\
\hat{Y}^{\eta,N}_{t}\in(-\gamma x_{0};0),\leavevmode\nobreak\
\hat{Z}^{\eta,N}_{t}\in(0;\gamma x_{0})\text{ for all }t\in(0;T)\Big{]}=1.$

We are interested in the dynamics of the optimal portfolio processes for small
instantaneous price impact. We address this problem within a factor model
where the cost coefficients $\lambda$ and $\rho$ are driven by an Itô
diffusion, which is given by the unique strong solution to the SDE

$\displaystyle
d\chi_{t}=\mu(t,\chi_{t})\mathrm{d}t+\sigma(t,\chi_{t})\mathrm{d}W_{t},\quad\chi_{0}=\overline{\chi}_{0}$

on $[0;T]$ with $\overline{\chi}_{0}\in\mathbb{R}^{n}$. We assume throughout
that the function
$(\mu,\sigma)\colon[0;T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}\times\mathbb{R}^{n\times
m}$ is bounded, measurable and uniformly Lipschitz continuous in the space
variable:

$\|(\mu,\sigma)(t,x_{1})-(\mu,\sigma)(t,x_{2})\|_{\infty}\leq
c\|x_{1}-x_{2}\|_{\infty}.$

######  Assumption 2.3.

The processes $\rho$ and $\lambda$ are of the form

$(\rho_{t},\lambda_{t})=f(t,\chi_{t})$

for some bounded $C^{1,2}$ function
$f=(f^{\rho},f^{\lambda})\colon[0;T]\times\mathbb{R}^{n}\to[0;\infty)^{2}$
with bounded derivatives. Moreover, the function $f^{\rho}$ is bounded away
from zero.

For convenience, we define the stochastic process
$\varphi=\\{\varphi_{t}\\}_{t\in[0;T]}$ by

$\varphi_{t}:=\sqrt{\lambda_{t}+2\gamma\rho_{t}},$

choose constants
$\underline{\rho},\overline{\rho},\underline{\varphi},\overline{\varphi},\overline{\lambda}\in(0;\infty)$
such that $\underline{\rho}\leq\rho\leq\overline{\rho}$,
$0\leq\lambda\leq\overline{\lambda},\underline{\varphi}\leq\varphi\leq\overline{\varphi}$.

In what follows, we heuristically argue that the processes $\hat{X}^{\eta,N}$
converge to a limit process $\hat{X}^{0}$ (independent of $N$) as $\eta\to 0$
and identify the limit $\hat{X}^{0}$. Since the ODE system (2.5) is not
defined for $\eta=0$, we cannot define the limiting process as the solution to
this system. Instead, we first identify the limits of the coefficients of the
ODE system and then derive candidate limits for the state processes in terms
of the limiting coefficients.

####  2.1.1 Convergence of the coefficient processes

In this section, we state the convergence results for the coefficient
processes $D^{\eta,N}$ and $E^{\eta,N}$ of the ODE system (2.5) as $\eta\to
0$. In particular, we prove that their limits $D^{0}$ and $E^{0}$ exist and
are driven by a common factor, which is given by the solution of a quadratic
BSDE.

###### Heuristics in the deterministic case.

Before proceeding to the limit result, we provide some heuristics for the
convergence. Assuming for simplicity that all coefficients are deterministic,
the dynamics of the coefficient processes satisfy

$\displaystyle\sqrt{\eta}\dot{D}^{\eta,N}_{t}=$
$\displaystyle(D^{\eta,N}_{t})^{2}-\lambda_{t}-\gamma\dot{B}^{\eta,N}_{t},$
$\displaystyle\sqrt{\eta}\dot{E}^{\eta,N}_{t}=$ $\displaystyle
2\sqrt{\eta}\rho_{t}E^{\eta,N}_{t}+\gamma(E^{\eta,N}_{t})^{2}-2\rho_{t}(1-B^{\eta,N}_{t})-\dot{B}^{\eta,N}_{t}.$

Letting $\eta\to 0$, we expect that

$\displaystyle 0=$
$\displaystyle(D^{0}_{t})^{2}-\lambda_{t}-\gamma\dot{B}^{0}_{t},$
$\displaystyle 0=$
$\displaystyle\gamma(E^{0}_{t})^{2}+2\rho_{t}(B^{0}_{t}-1)-\dot{B}^{0}_{t},$

that is, we expect that

$\displaystyle
D^{0}_{t}=\sqrt{\gamma\dot{B}^{0}_{t}+\lambda_{t}}\quad\mbox{and}\quad
E^{0}_{t}=\sqrt{\gamma^{-1}\big{(}\dot{B}^{0}_{t}+2\rho_{t}(1-B^{0}_{t})\big{)}}.$
(2.6)

Moreover, by the choice of the coefficients $D^{\eta,N}$ and $E^{\eta,N}$, we
expect that

$\displaystyle-\dot{B}^{0}_{t}=$
$\displaystyle-\rho_{t}B^{0}_{t}+D^{0}_{t}E^{0}_{t}.$ (2.7)

The three equalities combined yield

$\begin{split}\big{(}D^{0}_{t}+\gamma
E^{0}_{t}\big{)}^{2}&\leavevmode\nobreak\
=\gamma\dot{B}^{0}_{t}+\lambda_{t}+2\gamma
D^{0}_{t}E^{0}_{t}+\gamma\big{(}\dot{B}^{0}_{t}+2\rho_{t}(1-B^{0}_{t})\big{)}\\\
&\leavevmode\nobreak\
=(\varphi_{t})^{2}+2\gamma\big{(}\dot{B}^{0}_{t}+D^{0}_{t}E^{0}_{t}-\rho_{t}B^{0}_{t}\big{)}\\\
&\leavevmode\nobreak\ =(\varphi_{t})^{2}.\end{split}$ (2.8)

Plugging (2.7) and (2.8) back into (2.6) yields

$\displaystyle
D^{0}_{t}=(\varphi_{t})^{-1}(\lambda_{t}+\gamma\rho_{t}B^{0}_{t})\quad\mbox{and}\quad
E^{0}_{t}=(\varphi_{t})^{-1}\rho_{t}(2-B^{0}_{t}).$

Hence we expect $D^{0}$ and $E^{0}$ to be driven by $B^{0}$ and $B^{0}$ to
satisfy the ODE

$\displaystyle-\dot{B}^{0}_{t}=$
$\displaystyle-\rho_{t}B^{0}_{t}+D^{0}_{t}E^{0}_{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{\varphi_{t}^{2}}\Big{(}\gamma\big{(}\rho_{t}B^{0}_{t}\big{)}^{2}-2\lambda_{t}\rho_{t}\big{(}1-B^{0}_{t}\big{)}\Big{)}$

with terminal condition $B^{0}_{T}=1$ (because $B^{\eta,N}_{T}=1$). Our
heuristic also suggests that the limit processes are independent of the
liquidation requirement.

######  Example 2.4.

If $\lambda=C\rho$ for some constant $C\geq 0$, then the process $B^{0}$ can
be computed explicitly. If $C=0$, then
$B^{0}_{t}=2/({2+\int_{t}^{T}\rho_{s}\mathrm{d}s}).$ If $C>0$, then

$\displaystyle
B^{0}_{t}=-\frac{C}{\gamma}+\frac{1}{\gamma}\sqrt{C(C+2\gamma)}\coth\Bigg{(}\textup{arcoth}\bigg{(}\frac{C+\gamma}{\sqrt{C(C+2\gamma)}}\bigg{)}+\sqrt{\frac{C}{C+2\gamma}}\int_{t}^{T}\rho_{s}\mathrm{d}s\Bigg{)}.$

###### Convergence in the stochastic case.

The preceding heuristic suggests that the limiting coefficient processes are
driven by a solution to the BSDE corresponding to the above ODE for $B^{0}$.
The following lemma is proven in Section 3.1.2.

######  Lemma 2.5.

There exists a unique solution $(B^{0},Z^{0,B})$ in the space

$\mathcal{L}^{\infty}_{\mathcal{P}}(\Omega\times[0;T];(0;\infty))\times\mathcal{L}^{2}_{\mathcal{P}}(\Omega\times[0;T];\mathbb{R}^{m})$

to the BSDE

$\displaystyle-\mathrm{d}B^{0}_{t}=-\frac{1}{\varphi_{t}^{2}}\Big{(}\gamma\big{(}\rho_{t}B^{0}_{t}\big{)}^{2}-2\lambda_{t}\rho_{t}\big{(}1-B^{0}_{t}\big{)}\Big{)}\mathrm{d}t-Z^{0,B}_{t}\mathrm{d}W_{t},\quad
B^{0}_{T}=1.$ (2.9)

The process $B^{0}$ is bounded from above by $1$ and bounded from below by

$\displaystyle\underline{B}^{0}_{t}:=\exp\big{(}-\underline{\varphi}^{-2}\gamma\overline{\rho}^{2}(T-t)\big{)}.$
(2.10)

Moreover, there exists a uniformly continuous function
$h\colon[0;T]\times\mathbb{R}^{n}\to\mathbb{R}$ such that $B^{0}$ and
$\\{h(t,\chi_{t})\\}_{t\in[0;T]}$ are indistinguishable.

We prove below that the process $B^{\eta,N}$ converges to $B^{0}$ as $\eta\to
0$ and that $D^{\eta,N}$ and $E^{\eta,N}$ converge to the processes

$\begin{split}D^{0}_{t}:=(\varphi_{t})^{-1}(\lambda_{t}+\gamma\rho_{t}B^{0}_{t})\quad\mbox{and}\quad
E^{0}_{t}:=(\varphi_{t})^{-1}\rho_{t}(2-B^{0}_{t}),\end{split}$ (2.11)

respectively. In view of Lemma 2.5, these processes are well-defined and so
the dynamics of the process $B^{0}$ can be rewritten as

$-\mathrm{d}B^{0}_{t}=-\big{(}\rho_{t}B^{0}_{t}-D^{0}_{t}E^{0}_{t}\big{)}\mathrm{d}t-Z^{0,B}_{t}\mathrm{d}W_{t},\quad
B^{0}_{T}=1.$ (2.12)

Likewise, the BSDE for the process $B^{\eta,N}$ can be rewritten as

$\displaystyle-\mathrm{d}B^{\eta,N}_{t}=$
$\displaystyle-\Big{(}\rho_{t}B^{\eta,N}_{t}-D^{\eta,N}_{t}E^{\eta,N}_{t}\Big{)}\mathrm{d}t-Z^{\eta,N,B}_{t}\mathrm{d}W_{t},\quad
B^{\eta,N}_{T}=1.$ (2.13)

This suggests that the process $B^{\eta,N}$ converges to $B^{0}$ on the entire
interval $[0;T]$. By contrast, convergence of the processes $D^{\eta,N}$ and
$E^{\eta,N}$ can only be expected to hold on compact subintervals of $[0;T)$
because the terminal conditions of the limiting and the approximating
processes are different. Specifically, we have the following result. Its proof
is given in Section 3.2.1.

######  Proposition 2.6.

Let $b^{\eta,N}:=B^{\eta,N}-B^{0}$, $d^{\eta,N}:=D^{\eta,N}-D^{0}$ and
$e^{\eta,N}:=E^{\eta,N}-E^{0}$. Then, for all $\varepsilon>0$, there exists an
$\eta_{0}>0$ such that, for all $\eta\in(0;\eta_{0}]$ and for all
$N\in\mathcal{N}$,

$\displaystyle\mathbb{P}\bigg{[}$
$\displaystyle\sup_{t\in[0;T)}\max\big{(}\big{|}b^{\eta,N}_{t}\big{|},-d^{\eta,N}_{t},e^{\eta,N}_{t},-d^{\eta,N}_{t}-\gamma
e^{\eta,N}_{t}\big{)}\leq\varepsilon,$
$\displaystyle\quad\quad\inf_{t\in[0;T-\varepsilon]}\min\big{(}-d^{\eta,N}_{t},e^{\eta,N}_{t},-d^{\eta,N}_{t}-\gamma
e^{\eta,N}_{t}\big{)}\geq-\varepsilon\bigg{]}=1.$

####  2.1.2 Convergence of the state process

Having derived the limits of the coefficient processes, we can now
heuristically derive the limits of the processes $\hat{X}^{\eta,N}$,
$\hat{Y}^{\eta,N}$ and $\hat{Z}^{\eta,N}$, which we denote by $\hat{X}^{0}$,
$\hat{Y}^{0}$ and $\hat{Z}^{0}$, respectively.

##### The candidate processes.

Since $\hat{X}^{\eta,\infty}_{T}=0$ for all $\eta>0$, we expect that
$\hat{X}^{0}_{T}=0$. We prove in Lemma 3.11 that this convergence also holds
if $N$ is finite.666The proof heavily relies on the optimality of
$\hat{X}^{0}$ in the semimartingale portfolio liquidation model.

Assuming that the optimal trading strategy remains stable if $\eta\to 0$, the
ODE (2.5) suggests that the term
$D^{\eta,N}\hat{X}^{\eta,N}+E^{\eta,N}\hat{Y}^{\eta,N}$ is small for small
$\eta$ and hence that

$D^{0}_{t}\hat{X}^{0}_{t}=-E^{0}_{t}\hat{Y}^{0}_{t}\quad\mbox{on}\quad[0;T).$

We do not conjecture the above relation at the terminal time because the
convergence of $D^{\eta,N}$ and $E^{\eta,N}$ only holds on $[0;T)$. Assuming
that $\hat{Z}^{0}=\gamma\hat{X}^{0}-\hat{Y}^{0}$ on $[0;T)$, (2.8) implies

$\hat{Z}^{0}=-\frac{\varphi}{D^{0}}\hat{Y}^{0}\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{and hence}\leavevmode\nobreak\ \leavevmode\nobreak\
\hat{Y}^{0}=-\frac{D^{0}}{\varphi}\hat{Z}^{0}.$

On the other hand, multiplying the first equation in (2.5) by $\gamma$ and
subtracting the second and then taking the limit $\eta\to 0$ yields
$\partial_{t}\hat{Z}^{0}=\rho\hat{Y}^{0}$ and so

$\displaystyle\partial_{t}\hat{Z}^{0}_{t}=-\frac{\rho_{t}D^{0}_{t}}{\varphi_{t}}\hat{Z}^{0},\quad\hat{Z}^{0}_{0}=\gamma
x_{0}.$ (2.14)

We therefore expect that

$\displaystyle\hat{Z}^{0}_{t}=\gamma
x_{0}\exp\bigg{(}-\int^{t}_{0}\frac{\rho_{s}}{\varphi_{s}}D^{0}_{s}\mathrm{d}s\bigg{)},\quad
t\in[0;T].$

Since $\hat{X}^{0}_{T}=0$ and $\hat{Z}^{0}=\gamma\hat{X}^{0}-\hat{Y}^{0}$, we
finally expect that

$\displaystyle\hat{X}^{0}_{t}$
$\displaystyle=\mathbbm{1}_{[0;T)}(t)\frac{E^{0}_{t}}{\varphi_{t}}\hat{Z}^{0}_{t},$
$\displaystyle\hat{Y}^{0}_{t}$
$\displaystyle=-\mathbbm{1}_{[0;T)}(t)\frac{D^{0}_{t}}{\varphi_{t}}\hat{Z}^{0}_{t}-\mathbbm{1}_{\\{T\\}}(t)\hat{Z}^{0}_{T},\quad
t\in[0;T].$

Since $(\hat{X}^{0}_{0-},\hat{Y}^{0}_{0-})=(x_{0},0)$, this suggests that the
limiting state process jumps at the initial and the terminal time. In
particular, we cannot expect uniform convergence on $[0;T]$. We notice that
the limiting state process is of unbounded variation.

##### Convergence.

It remains to clarify in which sense the state processes converge. Contrary to
the convergence result stated in Proposition 2.6, we can only expect
convergence in probability because the state process follows a forward ODE
while the coefficient processes follow backward SDEs; see also Appendix B.2.
The following theorem establishes uniform convergence in probability on
compact subintervals on $(0;T)$ along with some “upper/lower convergence” at
the initial and terminal time. The proof is given in Section 3.2.2.

######  Theorem 2.7.

For all $\varepsilon>0$ and $\delta>0$, there exists an $\eta_{0}>0$ such
that, for all $\eta\in(0;\eta_{0}]$ and all $N\in\mathcal{N}$,

$\displaystyle\mathbb{P}\bigg{[}$
$\displaystyle\sup_{t\in[\varepsilon;T]}\max\big{(}\hat{X}^{\eta,N}_{t}-\hat{X}^{0}_{t},\hat{Y}^{\eta,N}_{t}-\hat{Y}^{0}_{t}\big{)}\leq\varepsilon,$
$\displaystyle\quad\inf_{t\in[0;T-\varepsilon]}\min\big{(}\hat{X}^{\eta,N}_{t}-\hat{X}^{0}_{t},\hat{Y}^{\eta,N}_{t}-\hat{Y}^{0}_{t}\big{)}\geq-\varepsilon\bigg{]}\geq
1-\delta.$

The preceding theorem does not provide a convergence result on the whole time
interval, due to the jumps of the limit processes at the initial and terminal
time. However, along with our results from Section 2.2, it allows us to prove
the convergence of the graphs of the state processes on the entire time
interval. The completed graph of a RCLL function
$X\colon\\{0-\\}\cup[0;T]\to\mathbb{R}$ with finitely many jumps is defined by

$\displaystyle G_{X}$
$\displaystyle:=\\{(t,x)\in[0;T]\times\mathbb{R}\,|\,x=X_{t}=X_{t-}\text{ or
}X_{t-}<X_{t},x\in[X_{t-};X_{t}]$ $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{ or
}X_{t-}>X_{t},x\in[X_{t};X_{t-}]\\}.$

The distance $d_{\infty}(G_{X},G_{Y})$ between two completed graphs is defined
as their Hausdorff distance:

$\displaystyle d_{\infty}(G_{X},G_{Y}):=\max\bigg{(}\sup_{p\in
G_{X}}\min_{q\in G_{Y}}\|p-q\|_{\infty},\sup_{q\in G_{Y}}\min_{p\in
G_{X}}\|p-q\|_{\infty}\bigg{)}\in[0;\infty].$

If strict liquidation is required, then Theorem 2.7 is sufficient to prove the
graph convergence of the state processes in the Hausdorff metric. If strict
liquidation is not required, an additional argument is needed to prove that
the terminal position converges to zero. The proof of the following
proposition is given in Section 3.2.3.

######  Proposition 2.8.

For all $\varepsilon>0$ and $\delta>0$, there exists an $\eta_{0}>0$ such
that, for all $\eta\in(0;\eta_{0}]$ and all $N\in\mathcal{N}$,

$\displaystyle\mathbb{P}\Big{[}d_{\infty}\big{(}G_{\hat{X}^{\eta,N}},G_{\hat{X}^{0}}\big{)}\leq\varepsilon\Big{]}\geq
1-\delta.$

Figure 1: One realisation of the optimal trading strategy in the stochastic
model introduced in [17] for different instantaneous impact factors $\eta$ and
its limit for $m=n=T=x_{0}=1$, $\gamma=3$, $\lambda\equiv 1$,
$\rho_{t}=1+0{.}9\sin(2{.}5W_{t})$.

###  2.2 Optimal liquidation with semimartingale strategies

In this section, we prove that the limit process $\hat{X}^{0}$ is the optimal
portfolio process in a trade execution model with semimartingale trading
strategies.

In our semimartingale model, a trading strategy is given by a triple
$\theta=(j^{+},j^{-},V)$ where $j^{+}$ and $j^{-}$ are real-valued, non-
decreasing pure jump processes and $V$ is a real-valued continuous Brownian
semimartingale starting in zero. The jump processes $j^{+}$ and $j^{-}$
describe the cumulative effects of buying, respectively selling large blocks
of shares while the continuous semimartingale $V$ describes the effect of
continuously trading small amounts of the stock. The portfolio process
$X^{\theta}=\\{X^{\theta}_{t}\\}_{t\in[0;T]}$ associated with a strategy
$\theta$ is then given by

$\displaystyle X^{\theta}_{t}=x_{0}+j^{+}_{t}-j^{-}_{t}+V_{t}.$

The associated price impact process, again given by (2.1), is denoted
$Y^{\theta}$. We note that $X^{\theta}$ and $Y^{\theta}$ are semimartingales.

We assume that strict liquidation is required and that the cost associated
with a trading strategy $\theta$ is given by

$\displaystyle
J^{0}(\theta):=\mathbb{E}\bigg{(}\frac{\gamma}{2}(j^{+}_{0}-j^{-}_{0})^{2}+\int_{(0;T]}\frac{1}{2}\big{(}Y^{\theta}_{s-}+Y^{\theta}_{s}\big{)}\mathrm{d}X^{\theta}_{s}+\int^{T}_{0}\frac{1}{2}\lambda_{s}\big{(}X^{\theta}_{s}\big{)}^{2}\mathrm{d}s+\frac{\gamma}{2}[V]_{T}\bigg{)}.$

The first term captures the cost of the initial block trade at time $t=0$. The
second term captures again the transient price impact cost; the third term
captures again market risk. The fourth term emerges as an additional cost term
when passing from discrete to continuous time, as shown in [1]. Moreover, in
the absence of this term, arbitrarily low costs can be achieved; see [1] for
details.

The cost functional is well defined under the following admissibility
condition.

######  Definition 2.9.

A trading strategy $\theta=(j^{+},j^{-},V)$ is called admissible if the
liquidation constraint $X^{\theta}_{T}=0$ holds, if $j^{\pm}$ is a RCLL,
predictable, real-valued, non-decreasing and square integrable pure jump
process, and $V$ is a continuous semimartingale starting in zero with

$\displaystyle\mathbb{E}\bigg{(}\max_{t\in[0;T]}|V_{t}|^{2}\bigg{)}<\infty.$
(2.15)

The set of all admissible trading strategies is denoted $\mathcal{A}^{0}$.

Our goal is now to solve the optimization problem

$\displaystyle\min_{\theta\in\mathcal{A}^{0}}J^{0}(\theta).$ (2.16)

To this end, we verify directly that the limit process $\hat{X}^{0}$ obtained
in the previous section is optimal. The results of Section 2.1 show that the
process has the following representation:

$\hat{X}^{0}_{t}=x_{0}-\hat{j}^{-}_{t}+\hat{V}_{t}$

where the jump process $\hat{j}^{-}$ and the continuous part are given by,
respectively

$\displaystyle\hat{j}^{-}_{t}$ $\displaystyle\leavevmode\nobreak\
:=x_{0}-\hat{X}^{0}_{0}+\mathbbm{1}_{\\{T\\}}(t)\hat{X}^{0}_{t-},$
$\displaystyle\hat{V}_{t}$ $\displaystyle\leavevmode\nobreak\
:=(\varphi_{t})^{-2}\rho_{t}\big{(}2-B^{0}_{t}\big{)}\hat{Z}^{0}_{t}-(\varphi_{0})^{-2}\rho_{0}\big{(}2-B^{0}_{0}\big{)}\hat{Z}^{0}_{0}.$

In view of Assumption 2.3 and because $B^{0}$ is a continuous semimartingale
and $\hat{Z}^{0}$ is differentiable, the process $\hat{V}$ is a continuous
semimartingale starting in zero. Hence the following holds.

######  Lemma 2.10.

The strategy $\hat{\theta}=(0,\hat{j}^{-},\hat{V})$ is admissible.

In order to prove that $\hat{\theta}$ is optimal, we approximate the cost and
the portfolio process associated with any strategy $\theta\in\mathcal{A}^{0}$
by the cost and portfolio processes corresponding to absolutely continuous
trading strategies. To this end, we first approximate the continuous
semimartingale part $V$ by differentiable processes. The proof of the
following Lemma is given in Section 4.1.

######  Lemma 2.11.

For all $\theta=(j^{+},j^{-},V)\in\mathcal{A}^{0}$ and for all
$\beta,\delta>0$, there exists a constant $\nu>0$ and an adapted and
continuous
$\dot{V}^{\beta,\nu}\colon\Omega\times[0;T]\to[-\frac{\beta}{\nu};\frac{\beta}{\nu}]$
such that

$\displaystyle\max_{t\in[0;T]}\bigg{|}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{|}\leq\max_{t\in[0;T]}|V_{t}|.$
(2.17)

and

$\displaystyle\mathbb{P}\bigg{[}\sup_{t\in[0;T]}\bigg{|}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s-V_{t}\bigg{|}\leq
3\beta\bigg{]}\geq 1-\delta.$ (2.18)

Next, we approximate the portfolio process $X^{\theta}$ by a portfolio process
associated with an absolutely continuous strategy. To this end, for all
$\theta\in\mathcal{A}^{0}$ and $\beta,\nu,\varepsilon>0$, we define the
integrable process

$\displaystyle\xi^{\theta,\beta,\nu,\varepsilon}_{t}:=\begin{cases}\frac{1}{\varepsilon}(j^{+}_{t}-j^{+}_{t-\varepsilon})-\frac{1}{\varepsilon}(j^{-}_{t}-j^{-}_{t-\varepsilon})+\dot{V}^{\beta,\nu}_{t},&t\leq
T-\varepsilon,\\\
-\frac{1}{\varepsilon}\bigg{(}x_{0}+\int^{T-\varepsilon}_{0}\tilde{\xi}^{\theta,\beta,\nu,\varepsilon}_{u}\mathrm{d}u\bigg{)}.&t>T-\varepsilon.\end{cases}$
(2.19)

In view of square-integrability of $j^{\pm}$, we see that
$\xi^{\theta,\beta,\nu,\varepsilon}$ belongs to $\mathcal{A}$. The
corresponding portfolio process is denoted $X^{\theta,\beta,\nu,\varepsilon}$.
The proof of the following Lemma is given in Section 4.2.

######  Lemma 2.12.

For all $\theta=(j^{+},j^{-},V)\in\mathcal{A}^{0}$ and for all $\delta>0$,
there exist $\beta,\nu,\varepsilon>0$ such that

$\displaystyle\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta,\beta,\nu,\varepsilon}_{t}-X^{\theta}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}\leq\delta.$

For all $\xi\in\mathcal{A}$ with $\int^{T}_{0}\xi_{s}\mathrm{d}s=-x_{0}$, we
can define $V^{\xi}_{t}:=\int^{t}_{0}\xi_{s}\mathrm{d}s$ such that
$(0,0,V^{\xi})\in\mathcal{A}^{0}$ and, for all $\eta>0$, we have

$\displaystyle
J^{\eta,\infty}(\xi)=J^{0}\big{(}0,0,V^{\xi}\big{)}+\frac{\eta}{2}\mathbb{E}\bigg{(}\int^{T}_{0}(\xi_{t})^{2}\mathrm{d}t\bigg{)}.$
(2.20)

The preceding lemma allows us to establish a cost estimate. The proof is given
in Section 4.3.

######  Lemma 2.13.

For all $\theta=(j^{+},j^{-},V)\in\mathcal{A}^{0}$, there exists a finite
constant $C(\theta)>0$ s.t. for all $\xi\in\mathcal{A}$ with
$\int^{T}_{0}\xi_{s}\mathrm{d}s=-x_{0}$,

$\displaystyle\big{|}J^{0}(\theta)-J^{0}\big{(}0,0,V^{\xi}\big{)}\big{|}\leq
C(\theta)\Bigg{(}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}}+\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}\Bigg{)}.$

The cost estimate allows us to establish the optimality of the trading
strategy $\hat{\theta}$.

######  Theorem 2.14.

The strategy $\hat{\theta}$ minimizes the cost functional $J^{0}$ over the set
$\mathcal{A}^{0}$.

###### Proof.

Let us assume to the contrary that $\hat{\theta}$ does not minimize the cost
functional $J^{0}$ over the set $\mathcal{A}^{0}$. Then, there exist a
strategy $\theta=(j^{+},j^{-},V)\in\mathcal{A}^{0}$ and a constant
$\delta\in(0;\infty)$ such that $J^{0}(\theta)+\delta\leq
J^{0}(\hat{\theta})$. Moreover, due to (2.20) and Lemma 2.13, for all
$\eta,\beta,\nu,\varepsilon\in(0;\infty)$, we have (for convecience, let
$p(x):=x+\sqrt{x}$)

$\displaystyle\leavevmode\nobreak\
J^{\eta,\infty}\big{(}\xi^{\theta,\beta,\nu,\varepsilon}\big{)}-J^{\eta,\infty}\big{(}\hat{\xi}^{\eta,\infty}\big{)}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
J^{\eta,\infty}\big{(}\xi^{\theta,\beta,\nu,\varepsilon}\big{)}-J^{0}(\theta)+J^{0}(\hat{\theta})-\delta-J^{0}\big{(}0,0,V^{\hat{\xi}^{\eta,\infty}}\big{)}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
-\delta+\frac{\eta}{2}\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}\xi^{\theta,\beta,\nu,\varepsilon}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}+\Big{|}J^{0}(\theta)-J^{0}\big{(}0,0,V^{\xi^{\theta,\beta,\nu,\varepsilon}}\big{)}\Big{|}+\Big{|}J^{0}\big{(}\hat{\theta}\big{)}-J^{0}\big{(}0,0,V^{\hat{\xi}^{\eta,\infty}}\big{)}\Big{|}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
-\delta+\frac{\eta}{2}\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}\xi^{\theta,\beta,\nu,\varepsilon}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}+C(\theta)\cdot
p\Bigg{(}\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\theta,\beta,\nu,\varepsilon}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}\Bigg{)}$
$\displaystyle\leavevmode\nobreak\ +C\big{(}\hat{\theta}\big{)}\cdot
p\Bigg{(}\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}\hat{X}^{0}_{t}-\hat{X}^{\eta,\infty}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}\Bigg{)}.$

According to Lemma 2.12, we can first choose
$\beta,\nu,\varepsilon\in(0;\infty)$ and then, due to
$|\hat{X}^{\eta,\infty}_{t}|\leq x_{0}$ (cf. Theorem 2.2), choose
$\eta\in(0;\infty)$ small enough according to Theorem 2.7 such that the sum is
negative. This contradicts to the optimality of $\hat{\xi}^{\eta,\infty}$ for
$J^{\eta,\infty}$. ∎

##  3 Proofs for Section 2.1

This section proves the results stated in Section 2.1. We start with a priori
estimates and regularity properties for the coefficient processes that specify
the optimal state processes.

###  3.1 A priori estimates and regularity properties

####  3.1.1 The case $\eta>0$

The following estimates have been established in [17, 21], except for the
upper bound on $E^{\eta,N}$ for finite $N$, which is stronger than the
corresponding one in [21]. It can be established using the same arguments as
in the proof of Proposition 3.2 in [17] noting that $D^{\eta,N}_{T}<\infty$ if
$N$ is finite.

######  Lemma 3.1.

Let

$\overline{\kappa}:=\sqrt{2\max(\overline{\lambda},\gamma\overline{\rho})}.$

For all $\eta\in(0;\infty)$, $N\in\mathcal{N}$ and $s,t\in[0;T)$ with $s\leq
t$, we have that

$\displaystyle\exp\big{(}-\overline{\rho}(T-s)\big{)}$ $\displaystyle\leq
B^{\eta,N}_{s}\leq 1,$ $\displaystyle 0$ $\displaystyle\leq
E^{\eta,N}_{s}\leq\gamma^{-1}\overline{\kappa}\tanh\Big{(}\sqrt{\eta^{-1}}\overline{\kappa}(T-s)\Big{)}\leq\gamma^{-1}\overline{\kappa},\quad$
$\displaystyle 0$
$\displaystyle<D^{\eta,N}_{s}\leq\overline{\kappa}\coth\big{(}\sqrt{\eta^{-1}}\overline{\kappa}(T-s)\big{)}\leq\overline{\kappa}\coth\big{(}\sqrt{\eta^{-1}}\overline{\kappa}(T-t)\big{)}.$

The preceding estimates allow us to prove that neither the optimal portfolio
process nor the corresponding spread process change sign.

###### Proof of Theorem 2.2.

Let us put $V(t):=\hat{X}^{\eta,N}_{t}\cdot\hat{Y}^{\eta,N}_{t}$. Then
$V(0)=0$ and $V$ satisfies the ODE

$\displaystyle\dot{V}(t)=$
$\displaystyle-\Big{(}\sqrt{\eta^{-1}}\big{(}D^{\eta,N}_{t}+\gamma
E^{\eta,N}_{t}\big{)}+\rho_{t}\Big{)}V(t)-\sqrt{\eta^{-1}}\Big{(}E^{\eta,N}_{t}\big{(}\hat{Y}^{\eta,N}_{t}\big{)}^{2}+\gamma
D^{\eta,N}_{t}\big{(}\hat{X}^{\eta,N}_{t}\big{)}^{2}\Big{)}$

As a result,

$\displaystyle V(t)=$
$\displaystyle-\exp\bigg{(}-\int^{t}_{0}\Big{(}\sqrt{\eta^{-1}}\big{(}D^{\eta,N}_{u}+\gamma
E^{\eta,N}_{u}\big{)}+\rho_{u}\big{)}\mathrm{d}u\bigg{)}\cdot$
$\displaystyle\qquad\int^{t}_{0}\sqrt{\eta^{-1}}\Big{(}E^{\eta,N}_{s}\big{(}\hat{Y}^{\eta,N}_{s}\big{)}^{2}+\gamma
D^{\eta,N}_{s}\big{(}\hat{X}^{\eta,N}_{s}\big{)}^{2}\Big{)}\exp\bigg{(}\int_{0}^{s}\Big{(}\sqrt{\eta^{-1}}\big{(}D^{\eta,N}_{u}+\gamma
E^{\eta,N}_{u}\big{)}+\rho_{u}\Big{)}\mathrm{d}u\bigg{)}\mathrm{d}s.$

In view of Lemma 3.1, this shows that $V(t)<0$ on $(0;T)$. Hence, strict
positivity of $\hat{X}^{\eta,N}_{0}$ yields

$\displaystyle\mathbb{P}\Big{[}\hat{X}^{\eta,N}_{t}>0,\hat{Y}^{\eta,N}_{t}<0\text{
for all }t\in(0;T)\Big{]}=1.$

Thus, the definition of $\hat{Z}^{\eta,N}$ along with (2.5) yields
$\partial_{t}\hat{Z}^{\eta,N}_{t}=\rho_{t}\hat{Y}^{\eta,N}_{t}<0$ on $(0;T)$.
Moreover,

$\displaystyle\hat{X}^{\eta,N}_{t}<\gamma^{-1}\hat{Z}^{\eta,N}_{t}\leq\gamma^{-1}\hat{Z}^{\eta,N}_{0}=x_{0}\quad\mbox{and}\quad\hat{Y}^{\eta,N}_{t}>-\hat{Z}^{\eta,N}_{t}\geq-\gamma
x_{0}.$

∎

Next, we prove that the process $B^{\eta,N}$ satisfies an $L^{1}$ uniform
continuity property. We refer to Appendix A for a discussion of general
regularity properties of stochastic processes.

######  Lemma 3.2.

The process $B^{\eta,N}$ satisfies Condition C.1 stated in Appendix A on
$[0;T]$ uniformly in $N\in\mathcal{N}$ and $\eta\in(0;H],$ for all
$H\in(0;\infty)$.

###### Proof.

Let $\varepsilon,\varepsilon_{1}>0$ and $s>T-\varepsilon_{1}$. By Lemma 3.1,
if $\varepsilon_{1}$ is small enough,

$\displaystyle\big{|}\mathbb{E}\big{(}V\big{(}B^{\eta,N}_{\tau}-B^{\eta,N}_{s}\big{)}\big{)}\big{|}\leq\big{(}1-\exp\big{(}-\overline{\rho}\varepsilon_{1}\big{)}\big{)}\mathbb{E}(|V|)\leq\varepsilon\mathbb{E}(|V|).$

If $s\leq T-\varepsilon_{1}$, the assertion follows from the integral
representation (2.13) along with the estimates established in Lemma 3.1 using
that the stochastic integral in (2.13) is a martingale on
$[0;T-\varepsilon_{1}]$777We emphasize that $Z^{\eta,N,B}$ is possibly defined
only on $[0;T-]$ and so the stochastic integral may be a martingale only away
from the terminal time. ∎

####  3.1.2 The case $\eta=0$

We are now going to establish a priori estimates on the candidate limiting
coefficient processes. First, we show that Assumption 2.3 directly implies the
following regularity result for the parameter processes:

######  Lemma 3.3.

The processes $\chi$, $\rho$, $\lambda$, $\varphi$ and $\varphi^{-1}$ satisfy
Condition C.2 introduced in Appendix A.

###### Proof.

$\chi$ satisfies Condition C.2 due to Lemma A.8, Lemma A.2 and Lemma A.4. The
rest immediately follows by Lemma A.7. ∎

We are now ready to prove that the process $B^{0}$ is well-defined.

###### Proof of Lemma 2.5.

The existence result follows from a standard argument. In fact, it is well
known that, for any $\overline{b}\in[1;\infty)$, the BSDE

$\displaystyle-\mathrm{d}B^{\overline{b}}_{t}=\psi^{\overline{b}}\big{(}t,\chi_{t},B^{\overline{b}}_{t}\big{)}\mathrm{d}t-Z^{\overline{b}}_{t}\mathrm{d}W_{t},\quad
B^{\overline{b}}_{T}=1$ (3.1)

with Lipschitz continuous driver

$\displaystyle\psi^{\overline{b}}(t,\chi,b):=-\frac{1}{f^{\varphi}(t,\chi)^{2}}\bigg{(}\gamma
f^{\rho}(t,\chi)^{2}\big{(}\overline{b}^{2}\wedge
b^{2}\big{)}-2f^{\lambda}(t,\chi)f^{\rho}(t,\chi)\Big{(}1-\big{(}0\vee(b\wedge\overline{b})\big{)}\Big{)}\bigg{)}$

has a unique solution
$(B^{\overline{b}},Z^{\overline{b}})\in\mathcal{L}_{\mathcal{P}}^{2}(\Omega\times[0;T];\mathbb{R}\times\mathbb{R}^{m})$
(cf. Theorem A.10). Let us then define the functions
$\underline{\phi},\overline{\phi}\colon\Omega\times[0;T]\times\mathbb{R}\times\mathbb{R}^{m}\to\mathbb{R}$
by

$\underline{\phi}(t,b,z):=-\underline{\varphi}^{-2}\gamma\overline{\rho}^{2}b\quad\mbox{and}\quad\overline{\phi}(t,b,z):=0.$

By definition, $(\underline{B}^{0},0)$ is the unique solution to the BSDE with
driver $\underline{\phi}$ and terminal condition $1$, where the lower bound
$\underline{B}^{0}$ on the process $B^{0}$ was defined in (2.10). Likewise,
$(1,0)$ is the unique solution of the BSDE with driver $\overline{\phi}$ and
the same terminal condition. Since

$\psi^{\overline{b}}\big{(}t,\chi_{t},\underline{B}^{0}_{t}\big{)}>\underline{\phi}\big{(}t,\underline{B}^{0}_{t},0\big{)}\quad\mbox{and}\quad\psi^{\overline{b}}(t,\chi_{t},1)<\overline{\phi}(t,1,0),$

the standard comparison principle for BSDEs with Lipschitz continuous drivers
yields

$\underline{B}^{0}\leq B^{\overline{b}}\leq 1.$

This proves that $(B^{1},Z^{1})$ is the desired unique bounded solution to the
BSDE (2.9).

The second assertion follows Theorem A.10 applied to the BSDE (3.1) for
$\overline{b}=1$. ∎

Having established the existence of the process $B^{0}$, the processes $D^{0}$
and $E^{0}$ are well-defined. The following lemma establishes estimates and
regularity properties for $D^{0}$ and $E^{0}$.

######  Lemma 3.4.

The following a priori estimates hold:

$\displaystyle\frac{\gamma\underline{\rho}}{\overline{\varphi}}\exp\bigg{(}-\frac{\gamma\overline{\rho}^{2}}{\underline{\varphi}^{2}}T\bigg{)}$
$\displaystyle\leq
D^{0}_{t}\leq\underline{\varphi}^{-1}\big{(}\overline{\lambda}+\gamma\overline{\rho}\big{)},$
$\displaystyle\overline{\varphi}^{-1}\underline{\rho}$ $\displaystyle\leq
E^{0}_{t}\leq 2\underline{\varphi}^{-1}\overline{\rho}.$

Moreover, the processes $D^{0}$ and $E^{0}$ satisfy Condition C.2.

###### Proof.

The a priori estimates can be obtained by plugging the bounds on $B^{0}$ (cf.
Lemma 2.5) into the definitions of $D^{0}$ and $E^{0}$ given in (2.11).
Moreover, if we denote by $h$ the function derived from Lemma 2.5, then for
all $t\in[0;T]$,

$\displaystyle D^{0}_{t}=\bigg{(}\frac{f^{\lambda}+\gamma\cdot f^{\rho}\cdot
h}{\sqrt{f^{\lambda}+2\gamma\cdot f^{\rho}}}\bigg{)}(t,\chi_{t}),\quad
E^{0}_{t}=\bigg{(}\frac{f^{\rho}\cdot(2-h)}{\sqrt{f^{\lambda}+2\gamma\cdot
f^{\rho}}}\bigg{)}(t,\chi_{t}).$

In view of Assumption 2.3, the processes $D^{0}$ and $E^{0}$ can be
represented as uniformly continuous functions of the factor process $\chi$ and
hence the assertion follows from Lemma 3.3 and Lemma A.7. ∎

The next lemma can be viewed as the analogue to Theorem 2.2 in the case
$\eta=0$.

######  Lemma 3.5.

It holds

$\displaystyle\mathbb{P}\big{[}\hat{X}^{0}_{t}\in(0;x_{0}),\hat{Y}^{0}_{t}\in(-\gamma
x_{0};0)\text{ for all }t\in(0;T)\big{]}=1.$

###### Proof.

Due to Lemma 3.4, $D^{0}$ is positive and hence, for $t\in(0;T)$,

$\displaystyle 0<\hat{Z}^{0}_{t}<x_{0}\gamma.$ (3.2)

Since $D^{0}+\gamma E^{0}=\varphi$ on $[0;T)$, it thus follows from Lemma 3.4
that, for all $t\in(0;T)$,

$\displaystyle 0<\hat{X}^{0}_{t}$
$\displaystyle=\frac{\varphi_{t}-D^{0}_{t}}{\gamma\varphi_{t}}\hat{Z}^{0}_{t}<\gamma^{-1}\hat{Z}^{0}_{t}<x_{0},$
$\displaystyle 0>\hat{Y}^{0}_{t}$ $\displaystyle=-\frac{\varphi_{t}-\gamma
E^{0}_{t}}{\varphi_{t}}\hat{Z}^{0}_{t}>-\hat{Z}^{0}_{t}>-\gamma x_{0}.$

∎

###  3.2 Proof of the convergence results

In this section, we prove our main convergence results. We start with the
convergence of the coefficients of the ODE system (2.5). Subsequently, we
prove that the convergence of the coefficients yields convergence of the state
process.

####  3.2.1 Proof of Proposition 2.6

The proof of Proposition 2.6 is split into a series of lemmas. In a first
step, we establish the convergence as $\eta\to 0$ of the auxiliary processes

$\displaystyle F^{\eta,N}_{t}:=D^{\eta,N}_{t}+\gamma
E^{\eta,N}_{t}\quad\mbox{to}\quad F^{0}:=\varphi$

and

$\displaystyle G^{\eta,N}:=\rho B^{\eta,N}+\varphi
E^{\eta,N}\quad\mbox{to}\quad G^{0}:=\rho B^{0}+\varphi E^{0}=2\rho.$

On $[0;T)$, the processes $F^{\eta,N}$ and $G^{\eta,N}$ satisfy the dynamics

$\begin{split}\mathrm{d}F^{\eta,N}_{t}=&\sqrt{\eta^{-1}}\Big{(}\big{(}F^{\eta,N}_{t}\big{)}^{2}-(\varphi_{t})^{2}\Big{)}\mathrm{d}t+2\gamma\rho_{t}E^{\eta,N}_{t}\mathrm{d}t\\\
&\quad+\sqrt{\eta^{-1}}\big{(}Z^{\eta,N,A}_{t}-2\gamma
Z^{\eta,N,B}_{t}+\gamma^{2}Z^{\eta,N,C}_{t}\big{)}\mathrm{d}W_{t},\end{split}$
(3.3)

respectively,

$\begin{split}-\mathrm{d}\big{(}\varphi^{-1}G^{\eta,N}\big{)}_{t}=&\sqrt{\eta^{-1}}\Big{(}-G^{\eta,N}_{t}+2\rho_{t}-E^{\eta,N}_{t}\big{(}F^{\eta,N}_{t}-\varphi_{t}\big{)}\Big{)}\mathrm{d}t-2\rho_{t}E^{\eta,N}_{t}\mathrm{d}t\\\
&\quad-\mathrm{d}\big{(}\varphi^{-1}\rho
B^{\eta,N}\big{)}_{t}+\sqrt{\eta^{-1}}\big{(}Z^{\eta,N,B}_{t}-\gamma
Z^{\eta,N,C}_{t}\big{)}\mathrm{d}W_{t}.\end{split}$ (3.4)

A general convergence result for integral equations of the above form is
established in Appendix B.1.2. It allows us to prove the following two lemmas.

######  Lemma 3.6.

Let $f^{\eta,N}:=F^{\eta,N}-\varphi$. For all $\varepsilon>0$, there exists an
$\eta_{0}>0$ such that, for all $\eta\in(0;\eta_{0}]$ and all
$N\in\mathcal{N}$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{s\in[0;T-\varepsilon]}f^{\eta,N}_{s}\leq\varepsilon,\inf_{s\in[0;T)}f^{\eta,N}_{s}\geq-\varepsilon\bigg{]}=1.$

###### Proof.

For every $N\in\mathcal{N}$, we apply Lemma B.4 to $P^{\eta}:=F^{\eta,N}$ with

$\psi:=1,\leavevmode\nobreak\ \leavevmode\nobreak\ a(x,y):=(x\vee
0)^{2}-y^{2},\leavevmode\nobreak\ \leavevmode\nobreak\
P^{0}:=\varphi,\leavevmode\nobreak\ \leavevmode\nobreak\
q^{\eta}:=0,\leavevmode\nobreak\ \leavevmode\nobreak\
L^{\eta}_{t}:=2\gamma\int^{t}_{0}\rho_{s}E^{\eta,N}_{s}\mathrm{d}s$

noticing that $\varepsilon\mapsto\eta_{0}=\eta_{0}(\varepsilon)$ is
independent of $N$. Assumption B.3 i) is satisfied with
$\overline{P}:=\overline{\varphi}+3\overline{\kappa}$, due to Lemma 3.1. If
$N=\infty$, then Assumption B.3 ii) a) is satisfied due to Theorem 2.1 and due
to the a priori estimates derived in Lemma 3.1 and Lemma 3.4. If $N<\infty$
and $\eta\leq\overline{\varphi}^{-2}$, then

$\displaystyle
f^{\eta,N}_{T}=D^{\eta,N}_{T}+\gamma\underbrace{E^{\eta,N}_{T}}_{=0}-D^{0}_{T}-\gamma
E^{0}_{T}=\sqrt{\eta^{-1}}(N-\gamma)-\varphi_{T}\geq\sqrt{\eta^{-1}}-\overline{\varphi}>0,$

which shows Assumption B.3 ii) b). Assumption B.3 iv) follows by direct
computation using $\overline{\varepsilon}:=\underline{\varphi}/2$ and
$\beta:=\underline{\varphi}/2$. Assumption B.3 v) follows from Lemma 3.3 and
Lemma 3.1. ∎

It is not difficult to show that similar arguments as those used to prove the
converegence of $f^{\eta,N}$ can be applied to $-g^{\eta,N}$. As a result, the
intervals $[0;T)$ and $[0;T-\varepsilon]$ in the statement of the convergence
result for $g^{\eta,N}$ need to be swapped.

######  Lemma 3.7.

Let $g^{\eta,N}:=G^{\eta,N}-2\rho$. For all $\varepsilon>0$, there exists an
$\eta_{0}>0$ such that, for all $\eta\in(0;\eta_{0}]$ and all
$N\in\mathcal{N}$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{s\in[0;T)}g^{\eta,N}_{s}\leq\varepsilon,\inf_{s\in[0;T-\varepsilon]}g^{\eta,N}_{s}\geq-\varepsilon\bigg{]}=1.$

We are now going to prove the almost sure convergence to zero of the process
$b^{\eta,N}:=B^{\eta,N}-B^{0}$. To this end, we first observe that
$D^{\eta,N}=F^{\eta,N}-\gamma E^{\eta,N}$ and $\varphi
E^{\eta,N}=G^{\eta,N}-\rho B^{\eta,N}$ yields

$\displaystyle-D^{\eta,N}E^{\eta,N}=-f^{\eta,N}E^{\eta,N}-G^{\eta,N}+\rho
B^{\eta,N}+\gamma\varphi^{-2}\big{(}G^{\eta,N}-\rho B^{\eta,N}\big{)}^{2}.$

Plugging this into (2.13) shows that

$\displaystyle\mathrm{d}B^{\eta,N}_{t}=\big{(}-G^{\eta,N}_{t}-f^{\eta,N}_{t}E^{\eta,N}_{t}+2\rho_{t}B^{\eta,N}_{t}+\gamma(\varphi_{t})^{-2}\big{(}G^{\eta,N}_{t}-\rho_{t}B^{\eta,N}_{t}\big{)}^{2}\big{)}\mathrm{d}t+Z^{\eta,N,B}_{t}\mathrm{d}W_{t}$

on $[0;T)$. Performing an analogous computation for $B^{0}$ and subtracting
the two equations yields

$\displaystyle\mathrm{d}b^{\eta,N}_{t}=$
$\displaystyle\Big{(}\big{(}b^{\eta,N}_{t}\big{)}^{2}\cdot\gamma(\varphi_{t})^{-2}(\rho_{t})^{2}+b^{\eta,N}_{t}\big{(}2\rho_{t}-2\gamma(\varphi_{t})^{-2}\rho_{t}\big{(}G^{0}_{t}-\rho_{t}B^{0}_{t}\big{)}\big{)}-f^{\eta,N}_{t}E^{\eta,N}_{t}-g^{\eta,N}_{t}$
$\displaystyle+\gamma(\varphi_{t})^{-2}g^{\eta,N}_{t}\big{(}g^{\eta,N}_{t}+2G^{0}_{t}-\rho_{t}\big{(}2b^{\eta,N}_{t}+2B^{0}_{t}\big{)}\big{)}\Big{)}\mathrm{d}t+\big{(}Z^{\eta,N,B}_{t}-Z^{0,B}_{t}\big{)}\mathrm{d}W_{t}$

on $[0;T)$. Using that

$\displaystyle
2\rho_{t}-2\gamma(\varphi_{t})^{-2}\rho_{t}\big{(}G^{0}_{t}-\rho_{t}B^{0}_{t}\big{)}$
$\displaystyle=2\rho_{t}(\varphi_{t})^{-1}D^{0}_{t},$ $\displaystyle
g^{\eta,N}_{t}+2G^{0}_{t}-\rho_{t}\big{(}2b^{\eta,N}_{t}+2B^{0}_{t}\big{)}$
$\displaystyle=-g^{\eta,N}_{t}+2\varphi_{t}E^{\eta,N}_{t}$

shows that

$\displaystyle\mathrm{d}b^{\eta,N}_{t}=$
$\displaystyle\Big{(}\big{(}b^{\eta,N}_{t}\big{)}^{2}\cdot\gamma(\varphi_{t})^{-2}(\rho_{t})^{2}+b^{\eta,N}_{t}\cdot
2(\varphi_{t})^{-1}\rho_{t}D^{0}_{t}-f^{\eta,N}_{t}E^{\eta,N}_{t}$
$\displaystyle+g^{\eta,N}_{t}\big{(}-1-\gamma(\varphi_{t})^{-2}g^{\eta,N}_{t}+2\gamma(\varphi_{t})^{-1}E^{\eta,N}_{t}\big{)}\Big{)}\mathrm{d}t+\big{(}Z^{\eta,N,B}_{t}-Z^{0,B}_{t}\big{)}\mathrm{d}W_{t}$
(3.5)

on $[0;T)$. This BSDE is different from (3.3) and (3.4). We apply Lemma B.2 to
prove the following result.

######  Lemma 3.8.

For all $\varepsilon>0$, there exists an $\eta_{0}>0$ such that, for all
$\eta\in(0;\eta_{0}]$ and all $N\in\mathcal{N}$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{s\in[0;T]}\big{|}b^{\eta,N}_{s}\big{|}\leq\varepsilon\bigg{]}=1.$

###### Proof.

For every $N\in\mathcal{N}$, we apply Lemma B.2 with

$\displaystyle a(t,b)$
$\displaystyle:=b^{2}\cdot\gamma(\varphi_{t})^{-2}(\rho_{t})^{2}+b\cdot
2(\varphi_{t})^{-1}\rho_{t}D^{0}_{t},$ $\displaystyle q^{\eta}_{t}$
$\displaystyle:=-f^{\eta,N}_{t}E^{\eta,N}_{t}+g^{\eta,N}_{t}\big{(}2\gamma(\varphi_{t})^{-1}E^{\eta,N}_{t}-1-\gamma(\varphi_{t})^{-2}g^{\eta,N}_{t}\big{)}.$

Assumption B.1 i) follows from the a priori estimates on $B^{\eta,N}$ (Lemma
3.1), $B^{0}$ (Lemma 2.5) and $D^{0}$ (Lemma 3.4). Assumption B.1 ii) follows
from the a priori estimates on $B^{\eta,N}$ and $B^{0}$, where the mapping
$\varepsilon\mapsto\delta$ is independent of $N$. Assumption B.1 iii) follows
from the same estimates and Lemma 3.6 and Lemma 3.7, where the choice of
$\eta_{1}$ is independent of $N$. Assumption B.1 iv) is satisfied because we
can choose $\varepsilon_{1}>0$ small enough s.t.

$-\varepsilon_{1}\gamma(\varphi)^{-1}\rho+2D^{0}\geq 0.$

∎

####  3.2.2 Proof of Theorem 2.7

First we need to prove an auxiliary result:

######  Lemma 3.9.

Let $I\subset\mathbb{R}$ be a compact interval and $Y=\\{Y_{t}\\}_{t\in I}$ a
continuous, adapted, $\mathbb{R}^{d}$-valued stochastic process. Then the
modulus of continuity

$\sup_{s,t\in I,|s-t|\leq\nu}\|Y_{t}-Y_{s}\|_{\infty}$

converges to $0$ in probability as $\nu\to 0$.

###### Proof.

For all $\omega\in\Omega$, $Y(\omega)$ is uniformly continuous, hence the
modulus of continuity converges to $0$ as $\nu\to 0$ $\mathbb{P}$-a.s. This
implies convergence in probability, in particular. ∎

###### Proof of Theorem 2.7.

Convergence on compact subintervals of $[0;T)$ follows from Theorem B.8
applied to the ODEs

$\displaystyle\begin{split}\partial_{t}\hat{X}^{\eta,N}_{t}=&\leavevmode\nobreak\
-\sqrt{\eta^{-1}}F^{\eta,N}_{t}\bigg{(}\hat{X}^{\eta,N}_{t}-\frac{E^{\eta,N}_{t}}{F^{\eta,N}_{t}}\hat{Z}^{\eta,N}_{t}\bigg{)},\\\
\partial_{t}\hat{Z}^{\eta,N}_{t}=&\leavevmode\nobreak\
\gamma\rho_{t}\hat{X}^{\eta,N}_{t}-\rho_{t}\hat{Z}^{\eta,N}_{t}\end{split}$
(3.6)

and

$\displaystyle\begin{split}\hat{X}^{0}_{t}=\frac{E^{0}_{t}}{\varphi_{t}}\hat{Z}^{0}_{t},\quad\hat{Z}^{0}_{t}=\hat{Z}^{0}_{0}\exp\bigg{(}\int^{t}_{0}\bigg{(}\gamma\rho_{s}\frac{E^{0}_{s}}{\varphi_{s}}-\rho_{s}\bigg{)}\mathrm{d}s\bigg{)}\end{split}$

on $[0;T)$. In fact, Condition (B.12) follows from Proposition 2.6 and
Condition (B.13) follows from Theorem 2.2. Finally, $D^{0}>0$ (Lemma 3.4),
from which we deduce that
$x_{0}>\frac{E^{0}_{0}}{\varphi_{0}}\hat{Z}^{0}_{0}$.

Thus, by Theorem B.8, applied once for every $N\in\mathcal{N}$ and every
$\nu>0$, for all $\nu,\delta\in(0;\infty)$, there exists
$\eta_{1}=\eta_{1}(\nu,\delta)\in(0;\infty)$ such that, for all
$\eta\in(0;\eta_{1}]$ and all $N\in\mathcal{N}$, the set

$\displaystyle M_{0}^{\nu,\eta,N}:=$
$\displaystyle\bigg{\\{}\sup_{t\in[0;T-\nu]}\max\Big{(}\big{|}\hat{Z}^{0}_{t}-\hat{Z}^{\eta,N}_{t}\big{|},\hat{X}^{0}_{t}-\hat{X}^{\eta,N}_{t}\Big{)}\leq\nu,\inf_{t\in[\nu;T-\nu]}\big{(}\hat{X}^{0}_{t}-\hat{X}^{\eta,N}_{t}\big{)}\geq-\nu\bigg{\\}}$

satisfies

$\displaystyle\mathbb{P}\Big{(}M_{0}^{\nu,\eta,N}\Big{)}>1-\delta/2.$ (3.7)

Near the terminal time, we cannot use this theorem since the convergence
$\frac{E^{\eta,N}}{F^{\eta,N}}\to\frac{E^{0}}{\varphi}$ holds only on
$[0;T-]$. Instead, we apply increment bounds on intervals of the form
$[T-\nu;T]$ to prove that, for all $\eta>0$, $N\in\mathcal{N}$ and for all
sufficiently small

$\displaystyle\nu\in(0;\varepsilon/2]\quad\quad\text{ and
}\quad\quad\beta\in(0;\varepsilon/3],$ (3.8)

the set

$\displaystyle M_{1}^{\nu,\varepsilon,\eta,N}:=$
$\displaystyle\bigg{\\{}\inf_{t\in[T-\nu;T)}\big{(}\hat{X}^{0}_{t}-\hat{X}^{\eta,N}_{t}\big{)}\geq-\varepsilon\bigg{\\}}$

contains the set $M_{0}^{\nu,\eta,N}\cap M_{2}^{\beta,\nu}\cap
M_{3}^{\nu,\eta,N}$, where

$\displaystyle M_{2}^{\beta,\nu}:=$
$\displaystyle\bigg{\\{}\sup_{s,t\in[T-\nu;T)}|\varphi_{s}-\varphi_{t}|\vee\big{|}E^{0}_{s}-E^{0}_{t}\big{|}\vee\big{|}\hat{X}^{0}_{s}-\hat{X}^{0}_{t}\big{|}\leq\beta\bigg{\\}},$
$\displaystyle M_{3}^{\nu,\eta,N}:=$
$\displaystyle\bigg{\\{}\sup_{t\in[T-\nu;T)}e^{\eta,N}_{t}\vee\big{(}-f^{\eta,N}_{t}\big{)}\leq\nu\bigg{\\}}.$

The probability of the last two events can be made large, due to Lemma 3.9 and
Proposition 2.6: for all $\beta,\delta,\nu\in(0;\infty)$, there exist
$\nu_{0}(\beta,\delta),\eta_{2}(\nu)\in(0;\infty)$ such that

$\displaystyle\inf_{\nu\leq\nu_{0}(\beta,\delta)}\mathbb{P}(M_{2}^{\beta,\nu})>1-\delta/2,\quad\inf_{\eta\leq\eta_{2}(\nu)}\inf_{N\in\mathcal{N}}\mathbb{P}(M_{3}^{\nu,\eta,N})=1.$
(3.9)

In order to see that $M_{1}^{\nu,\varepsilon,\eta,N}\supseteq
M_{0}^{\nu,\eta,N}\cap M_{2}^{\beta,\nu}\cap M_{3}^{\nu,\eta,N}$, we assume to
the contrary that there exists

$\omega\in\left(M_{0}^{\nu,\eta,N}\cap M_{2}^{\beta,\nu}\cap
M_{3}^{\nu,\eta,N}\right)\backslash M_{1}^{\nu,\varepsilon,\eta,N}$

and show that, for any such $\omega$, there exists a time
$s(\omega)\in[T-\nu;T)$ such that
$\partial_{t}\hat{X}^{\eta,N}_{s(\omega)}(\omega)$ must be both non-negative
and strictly negative at the same time.

We start with the choice of $s(\omega)$. Since $\omega\notin
M_{1}^{\nu,\varepsilon,\eta,N}$, there exists $t\in[T-\nu;T)$ such that
$\hat{X}^{\eta,N}_{t}(\omega)-\hat{X}^{0}_{t}(\omega)>\varepsilon$. Since
$\omega\in M_{2}^{\beta,\nu}$ and due to (3.8), this yields that

$\hat{X}^{\eta,N}_{t}(\omega)>\hat{X}^{0}_{T-\nu}(\omega)+\frac{2\varepsilon}{3}.$
(3.10)

Since $\omega$ belongs to $M_{0}^{\nu,\eta,N}$ and due to (3.8), this implies
that

$\displaystyle\hat{X}^{\eta,N}_{T-\nu}(\omega)<\hat{X}^{0}_{T-\nu}(\omega)+\frac{2\varepsilon}{3}<\hat{X}^{\eta,N}_{t}(\omega).$
(3.11)
$\displaystyle\hat{X}^{\eta,N}_{s(\omega)}(\omega)=\hat{X}^{0}_{T-\nu}(\omega)+\frac{2\varepsilon}{3}.$
(3.12)

Due to minimality of $s(\omega)$, we have
$\partial_{t}\hat{X}^{\eta,N}_{s(\omega)}(\omega)\geq 0$.

We now show that this derivative must also be strictly negative. In fact, due
to (3.6),

$\displaystyle\sqrt{\eta}\partial_{t}\hat{X}^{\eta,N}_{s(\omega)}(\omega)=$
$\displaystyle-F^{\eta,N}_{s(\omega)}(\omega)\hat{X}^{\eta,N}_{s(\omega)}(\omega)+E^{\eta,N}_{s(\omega)}(\omega)\hat{Z}^{\eta,N}_{s(\omega)}(\omega).$

Since $\hat{Z}^{\eta,N}$ is non-increasing (Theorem 2.2), using (3.12) again,
the right hand side of the above equation can be bounded from above by

$\displaystyle-\varphi_{T-\nu}(\omega)\hat{X}^{0}_{T-\nu}(\omega)+(\varphi_{T-\nu}(\omega)-\varphi_{s(\omega)}(\omega))\hat{X}^{0}_{T-\nu}(\omega)-f^{\eta,N}_{s(\omega)}(\omega)\hat{X}^{0}_{T-\nu}(\omega)$
$\displaystyle-F^{\eta,N}_{s(\omega)}(\omega)\frac{2\varepsilon}{3}+E^{0}_{T-\nu}(\omega)\hat{Z}^{0}_{T-\nu}(\omega)+E^{0}_{T-\nu}(\omega)\big{(}\hat{Z}^{\eta,N}_{T-\nu}(\omega)-\hat{Z}^{0}_{T-\nu}(\omega)\big{)}$
$\displaystyle+\big{(}E^{0}_{s(\omega)}(\omega)-E^{0}_{T-\nu}(\omega)\big{)}\hat{Z}^{\eta,N}_{T-\nu}(\omega)+e^{\eta,N}_{s(\omega)}(\omega)\hat{Z}^{\eta,N}_{T-\nu}(\omega).$

Since $\omega\in M_{3}^{\nu,\eta,N}$, we have
$F^{\eta,N}_{s(\omega)}(\omega)=\varphi_{s(\omega)}(\omega)+f^{\eta,N}_{s(\omega)}(\omega)\geq\underline{\varphi}-\nu,$
which is strictly positive if $\nu<\underline{\varphi}$. Moreover,
$-\varphi\hat{X}^{0}+E^{0}\hat{Z}^{0}=0$ on $[0;T)$. Since $\omega\in
M_{0}^{\nu,\eta,N}\cap M_{2}^{\beta,\nu}\cap M_{3}^{\nu,\eta,N}$ and since
$\hat{X}^{0}$, $E^{0}$ and $\hat{Z}^{\eta,N}$ are bounded, this shows that
$\partial_{t}\hat{X}^{\eta,N}_{s(\omega)}(\omega)<0$ if $\nu$ and $\beta$ are
chosen small enough.

The convergence of $\hat{Y}^{\eta,N}-\hat{Y}^{0}$ on $[0;T-]$ follows from

$\displaystyle\hat{Y}^{\eta,N}-\hat{Y}^{0}=\gamma\big{(}\hat{X}^{\eta,N}-\hat{X}^{0}\big{)}-\big{(}\hat{Z}^{\eta,N}-\hat{Z}^{0}\big{)}$

and (3.7). The convergence on $[T-;T]$ follows from the fact that

$\displaystyle\sup_{s\in[T-\nu;T]}\Big{|}\big{(}\hat{Z}^{\eta,N}_{s}-\hat{Z}^{0}_{s}\big{)}-\big{(}\hat{Z}^{\eta,N}_{T-\nu}-\hat{Z}^{0}_{T-\nu}\big{)}\Big{|}$

can be made arbitrarily small by choosing $\nu$ small since
$\partial_{t}\hat{Z}^{\eta,N}=\rho\hat{Y}^{\eta,N}$, since
$\partial_{t}\hat{Z}^{0}=\rho\hat{Y}^{0}$ and because both $\hat{Y}^{\eta,N}$
and $\hat{Y}^{0}$ are bounded. ∎

####  3.2.3 Proof of Proposition 2.8

The proof of the graph convergence of the optimal portfolio processes in the
Hausdorff topology follows from Theorem 2.7 if strict liquidation is required.
If strict liquidation is not required, the results of Section 2.2 are required
to establish the assertion. This is not a circular argument since the proofs
of Section 2.2 only use the results of Section 2.1 concerning the liquidating
case $N=\infty$.

Since $\hat{\theta}=(0,\hat{j}^{-},\hat{V})$ is optimal in the model
introduced in Section 2.2 (cf. Theorem 2.14), it is easy to show that the
strategy $\hat{\theta}^{q}:=(\hat{j}^{+,q},\hat{j}^{-,q},\hat{V}^{q})$ is
admissible for every $q\in\mathbb{R}$ where $\hat{V}^{q}_{t}:=\hat{V}_{t}+qt$
and
$\hat{j}^{+,q}_{t}-\hat{j}^{-,q}_{t}=-\hat{j}^{-}_{t}-\mathbbm{1}_{\\{T\\}}(t)qT$.

The following lemma shows that we can express the cost term corresponding to
the transient price impact without Itô integrals. The proof is an immediate
consequence of Itô’s formula for semimartingales (Theorem II.32 in [32]) and
the fact that $\gamma^{2}[V]$ is equal to the continuous part of
$[Y^{\theta}]$.

######  Lemma 3.10.

For all $\theta=(j^{+},j^{-},V)\in\mathcal{A}^{0}$ and all $t\in[0;T]$, it
holds that

$\displaystyle\int_{(0;t]}\frac{1}{2}\big{(}Y^{\theta}_{s-}+Y^{\theta}_{s}\big{)}\mathrm{d}X^{\theta}_{s}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\frac{1}{2\gamma}\big{(}Y^{\theta}_{t}\big{)}^{2}-\frac{\gamma}{2}(j^{+}_{0}-j^{-}_{0})^{2}-\frac{\gamma}{2}[V]_{t}+\frac{1}{\gamma}\int^{t}_{0}\rho_{s}\big{(}Y^{\theta}_{s}\big{)}^{2}\mathrm{d}s.$

Using the previous lemma, it is not difficult to check that the mapping
$q\mapsto J^{0}(\hat{\theta}^{q})$ is differentiable and that

$\displaystyle\begin{split}0=\frac{\partial J^{0}(\hat{\theta}^{q})}{\partial
q}\bigg{|}_{q=0}=&\mathbb{E}\bigg{(}-\hat{Y}^{0}_{T}e^{-\int^{T}_{0}\rho_{u}\mathrm{d}u}\int^{T}_{0}t\rho_{t}e^{\int_{0}^{t}\rho_{u}\mathrm{d}u}\mathrm{d}t+\int^{T}_{0}2t\rho_{t}\hat{Y}^{0}_{t}\mathrm{d}t\\\
&\quad-\int^{T}_{0}2\rho_{t}\hat{Y}^{0}_{t}e^{-\int^{t}_{0}\rho_{u}\mathrm{d}u}\int^{t}_{0}s\rho_{s}e^{\int_{0}^{s}\rho_{u}\mathrm{d}u}\mathrm{d}s\mathrm{d}t+\int^{T}_{0}t\lambda_{t}\hat{X}^{0}_{t}\mathrm{d}t\bigg{)}\end{split}$
(3.13)

This allows us to prove that full liquidation is optimal if $\eta\to 0$ even
if it is not formally required.

######  Lemma 3.11.

We have that $\lim_{\eta\to
0}\sup_{N\in\mathcal{N}}\mathbb{E}(\hat{X}^{\eta,N}_{T})=0$.

###### Proof.

We assume to the contrary that $\limsup_{\eta\to
0}\sup_{N\in\mathcal{N}}\mathbb{E}(\hat{X}^{\eta,N}_{T})>0$ and prove that
this contradicts the optimality of $J^{\eta,N}(\hat{\xi}^{\eta,N})$. To this
end, we consider the admissible trading strategies

$\displaystyle\hat{\xi}^{\eta,N,q}_{t}:=\hat{\xi}^{\eta,N}_{t}+q,$

compute the derivative of the function $q\mapsto
J^{\eta,N}(\hat{\xi}^{\eta,N,q})$ and show that the derivative at $q=0$ does
not vanish for small $\eta$ if $\limsup_{\eta\to
0}\sup_{N\in\mathcal{N}}\mathbb{E}(\hat{X}^{\eta,N}_{T})>0$. Obviously,
$\hat{\xi}^{\eta,N,q}\in\mathcal{A}$ and

$\displaystyle\hat{X}^{\eta,N,q}_{t}:=X^{\hat{\xi}^{\eta,N,q}}_{t}=\hat{X}^{\eta,N}_{t}+qt.$

Straightforward computations lead to

$\displaystyle 0=\frac{\partial J^{\eta,N}(\hat{\xi}^{\eta,N,q})}{\partial
q}\bigg{|}_{q=0}=$
$\displaystyle\mathbb{E}\bigg{(}\eta\big{(}\hat{X}^{\eta,N}_{T}-x_{0}\big{)}-\hat{Y}^{\eta,N}_{T}e^{-\int^{T}_{0}\rho_{u}\mathrm{d}u}\int^{T}_{0}t\rho_{t}e^{\int_{0}^{t}\rho_{u}\mathrm{d}u}\mathrm{d}t+\int^{T}_{0}2t\rho_{t}\hat{Y}^{\eta,N}_{t}\mathrm{d}t$
$\displaystyle-\int^{T}_{0}2\rho_{t}\hat{Y}^{\eta,N}_{t}e^{-\int^{t}_{0}\rho_{u}\mathrm{d}u}\int^{t}_{0}s\rho_{s}e^{\int_{0}^{s}\rho_{u}\mathrm{d}u}\mathrm{d}s\mathrm{d}t+\int^{T}_{0}t\lambda_{t}\hat{X}^{\eta,N}_{t}\mathrm{d}t$
$\displaystyle+(N-\gamma)T\hat{X}^{\eta,N}_{T}+\gamma\hat{X}^{\eta,N}_{T}e^{-\int^{T}_{0}\rho_{u}\mathrm{d}u}\int^{T}_{0}t\rho_{t}e^{\int_{0}^{t}\rho_{u}\mathrm{d}u}\mathrm{d}t\bigg{)}$

Subtracting (3.13) yields

$\displaystyle 0=$ $\displaystyle\mathbb{E}\bigg{(}-\eta
x_{0}+\hat{X}^{\eta,N}_{T}(\eta+(N-\gamma)T)+\big{(}\hat{Z}^{\eta,N}_{T}-\hat{Z}^{0}_{T}\big{)}e^{-\int^{T}_{0}\rho_{u}\mathrm{d}u}\int^{T}_{0}t\rho_{t}e^{\int_{0}^{t}\rho_{u}\mathrm{d}u}\mathrm{d}t$
$\displaystyle+\int^{T}_{0}2t\rho_{t}\big{(}\hat{Y}^{\eta,N}_{t}-\hat{Y}^{0}_{t}\big{)}\mathrm{d}t-\int^{T}_{0}2\rho_{t}\big{(}\hat{Y}^{\eta,N}_{t}-\hat{Y}^{0}_{t}\big{)}e^{-\int^{t}_{0}\rho_{u}\mathrm{d}u}\int^{t}_{0}s\rho_{s}e^{\int_{0}^{s}\rho_{u}\mathrm{d}u}\mathrm{d}s\mathrm{d}t$
$\displaystyle+\int^{T}_{0}t\lambda_{t}\big{(}\hat{X}^{\eta,N}_{t}-\hat{X}^{0}_{t}\big{)}\mathrm{d}t\bigg{)}$
$\displaystyle\geq$ $\displaystyle-\eta
x_{0}+(\eta+(N-\gamma)T)\mathbb{E}\big{(}\hat{X}^{\eta,N}_{T}\big{)}-T^{2}\overline{\rho}e^{T\overline{\rho}}\mathbb{E}\Big{(}\big{|}\hat{Z}^{\eta,N}_{T}-\hat{Z}^{0}_{T}\big{|}\Big{)}$
$\displaystyle-2\big{(}T\overline{\rho}+\overline{\rho}^{2}T^{2}e^{T\overline{\rho}}\big{)}\mathbb{E}\bigg{(}\int^{T}_{0}\big{|}\hat{Y}^{\eta,N}_{t}-\hat{Y}^{0}_{t}\big{|}\mathrm{d}t\bigg{)}-T\overline{\lambda}\mathbb{E}\bigg{(}\int^{T}_{0}\big{|}\hat{X}^{\eta,N}_{t}-\hat{X}^{0}_{t}\big{|}\mathrm{d}t\bigg{)}$

In view of Theorem 2.2, Theorem 2.7, Lemma 3.5 and using
$\partial_{t}(\hat{Z}^{\eta,N}-\hat{Z}^{0})=\rho(\hat{Y}^{\eta,N}-\hat{Y}^{0})$,
the sum of the three last expected values is small uniformly in $N$ if $\eta$
is small. Hence, if $\limsup_{\eta\to
0}\sup_{N\in\mathcal{N}}\mathbb{E}(\hat{X}^{\eta,N}_{T})>0$, then the sum on
the right hand side of the above inequality is strictly positive when first
choosing $\eta_{0}>0$ small enough and then choosing
$(\tilde{\eta},\tilde{N})$ with $\tilde{\eta}\leq\eta_{0}$ such that

$\mathbb{E}\big{(}X^{\tilde{\eta},\tilde{N}}_{T}\big{)}\geq\frac{1}{2}\limsup_{\eta\to
0}\sup_{N\in\mathcal{N}}\mathbb{E}\big{(}\hat{X}^{\eta,N}_{T}\big{)}.$

. ∎

We are now ready to prove the convergence of the graph of $\hat{X}^{\eta,N}$
to the completed graph of $\hat{X}^{0}$ in the Hausdorff distance. To this
end, we have to bound the distance of each point of any of the graphs to the
other graph. In the inner interval $[\varepsilon;T-\varepsilon]$, it is enough
to consider $\hat{X}^{\eta,N}-\hat{X}^{0}$, which we have bounded by Theorem
2.7.

###### Proof of Proposition 2.8.

To prove that the probability of

$d_{\infty}\big{(}G_{\hat{X}^{\eta,N}},G_{\hat{X}^{0}}\big{)}\leq\varepsilon$

is large for small $\eta>0$, we need to prove that the distance of any point
$(t,x)$ from either $G_{\hat{X}^{\eta,N}(\omega)}$ or
$G_{\hat{X}^{0}(\omega)}$ to the respective other graph is small on a set of
large probability. To this end, we fix a small enough $\nu\in(0;\varepsilon)$.
If $t\in[\nu;T-\nu]$, this follows directly from Theorem 2.7. For
$t\in[0;\nu)\cup(T-\nu;T]$, we use Theorem 2.7 along with the facts that (i)
the completed graph of a discontinuous function contains the line segments
joining the values of the function at the points of discontinuity; (ii) the
increments of $\hat{X}^{0}$ are small in the sense of Lemma 3.9 and (iii)
$\lim_{\eta\to 0}\sup_{N\in\mathcal{N}}\mathbb{E}(\hat{X}^{\eta,N}_{T})=0$,
due to Lemma 3.11. For instance, let us consider an $\omega\in\Omega$ with

$\sup_{u\in[0;T-\nu]}\big{(}\hat{X}^{0}_{u}(\omega)-\hat{X}^{\eta,N}_{u}(\omega)\big{)}\leq\nu,$

and assume that $(t,x)\in G_{\hat{X}^{\eta,N}(\omega)}$ with $t<\nu$ and
$x<\hat{X}^{0}_{0}(\omega)-\nu$. Since $x=\hat{X}^{\eta,N}_{t}(\omega)$, the
mean value theorem yields an $s\in[0;t]$ s.t. $\hat{X}^{0}_{s}(\omega)=x+\nu$,
which proves that $(s,x+\nu)\in G_{\hat{X}^{0}(\omega)}$ and

$d_{\infty}((t,x),(s,x+\nu))=(t-s)+\nu\leq 2\nu.$

∎

##  4 Proofs for Section 2.2

###  4.1 Proof of Lemma 2.11

In order to prove Lemma 2.11 we first define, for all $\beta,\nu\in(0;\infty)$
and $x\in\mathbb{R}$,

$\displaystyle f^{\beta,\nu}(x):=\begin{cases}-\beta/\nu,&x\leq-\beta,\\\
x/\nu,&-\beta\leq x\leq\beta,\\\ \beta/\nu,&x\geq\beta.\end{cases}$

For any admissible strategy $(j^{+},j^{-},V)\in\mathcal{A}^{0}$, there exists
a unique pathwise differentiable, adapted stochastic process
$\\{\tilde{V}^{\beta,\nu}_{t}\\}_{t\in[0;T]}$ such that
$\tilde{V}^{\beta,\nu}_{0}=0$ and its time derivative satisfies

$\displaystyle\dot{V}^{\beta,\nu}_{t}(\omega)=f^{\beta,\nu}\big{(}V_{t}(\omega)-\tilde{V}^{\beta,\nu}_{t}(\omega)\big{)}.$

Now, (2.17) can easily be verified by the comparison principle: For all
$\varepsilon>0$ and all $t\in[0;T]$, we have

$\displaystyle
f^{\beta,\nu}\bigg{(}V_{t}-\bigg{(}\max_{s\in[0;T]}|V_{s}|+\varepsilon\bigg{)}\bigg{)}<0$

and hence, for all $\varepsilon>0$ and $t\in[0;T]$,

$\displaystyle\tilde{V}^{\beta,\nu}_{t}\leq\max_{s\in[0;T]}|V_{s}|+\varepsilon.$

Analogously, we can prove that
$\tilde{V}^{\beta,\nu}_{t}\geq-\max_{s\in[0;T]}|V_{s}|-\varepsilon$. Now, we
are going to prove (2.18). For $\beta,\nu\in(0;\infty)$, let

$\displaystyle
M:=\bigg{\\{}\sup_{s,t\in[0;T],|s-t|\leq\nu}\big{|}V_{s}-V_{t}\big{|}\leq\beta\bigg{\\}}.$

In view of Lemma 3.9, it is enough to prove that

$\displaystyle|V_{t}(\omega)-\tilde{V}^{\beta,\nu}_{t}(\omega)|<3\beta$

for all $\omega\in M$ and $t\in[0;T]$. In order to see this, let us assume to
the contrary that the statement is wrong. Then, by continuity, since
$V_{0}-\tilde{V}^{\beta,\nu}_{0}=0$, there exists some $\omega\in M$ and some
$t_{1}\in[0;T]$ such that

$\displaystyle|V_{t_{1}}(\omega)-\tilde{V}^{\beta,\nu}_{t_{1}}(\omega)|=3\beta.$

We choose the smallest such $t_{1}\in[0;T]$. Then, for each
$t\in[0\vee(t_{1}-\nu);t_{1}]$,

$\displaystyle|V_{t}(\omega)-\tilde{V}^{\beta,\nu}_{t}(\omega)|\geq$
$\displaystyle\leavevmode\nobreak\
|V_{t_{1}}(\omega)-\tilde{V}^{\beta,\nu}_{t_{1}}(\omega)|-|V_{t_{1}}(\omega)-V_{t}(\omega)|-|\tilde{V}^{\beta,\nu}_{t_{1}}(\omega)-\tilde{V}^{\beta,\nu}_{t}(\omega)|$
$\displaystyle\geq$ $\displaystyle\leavevmode\nobreak\
3\beta-\beta-(t_{1}-t)\frac{\beta}{\nu}$ $\displaystyle\geq$
$\displaystyle\leavevmode\nobreak\ \beta.$

Since $V_{0}-\tilde{V}^{\beta,\nu}_{0}=0$, we have $t_{1}>\nu$. Since
$V_{\cdot}(\omega)-\tilde{V}^{\beta,\nu}_{\cdot}(\omega)$ is continuous and
has no roots in $[t_{1}-\nu;t_{1}]$, it does not change sign on this interval.
We may hence w.l.o.g. assume that
$V_{t_{1}}(\omega)-\tilde{V}^{\beta,\nu}_{t_{1}}(\omega)=3\beta$. This implies
that

$\displaystyle V_{t}(\omega)-\tilde{V}^{\beta,\nu}_{t}(\omega)\geq\beta$

for all $t\in[t_{1}-\nu;t_{1}]$. By definition, for all those $t$,
$\dot{V}^{\beta,\nu}_{t}(\omega)=\beta/\nu$. This, however, contradicts the
minimality of $t_{1}$ as

$\displaystyle
V_{t_{1}-\nu}(\omega)-\tilde{V}^{\beta,\nu}_{t_{1}-\nu}(\omega)=$
$\displaystyle
V_{t_{1}-\nu}(\omega)-V_{t_{1}}(\omega)+V_{t_{1}}(\omega)-\tilde{V}^{\beta,\nu}_{t_{1}}(\omega)+\tilde{V}^{\beta,\nu}_{t_{1}}(\omega)-\tilde{V}^{\beta,\nu}_{t_{1}-\nu}(\omega)$
$\displaystyle\geq$
$\displaystyle-\beta+3\beta+\int_{t_{1}-\nu}^{t_{1}}\underbrace{\dot{V}^{\beta,\nu}_{s}}_{=\beta/\nu}\mathrm{d}s=3\beta.$

This finishes the proof of Lemma 2.11. $\Box$

###  4.2 Proof of Lemma 2.12

The proof of Lemma 2.12 requires the following result on the jump processes.

######  Lemma 4.1.

Let $\\{j_{t}\\}_{t\in[0;T]}$ be a $[0;\infty)$-valued, non-decreasing and
progressively measurable stochastic process with
$\mathbb{E}((j_{T})^{2})<\infty$. Then (putting $j_{t}=0$ for $t<0$)

$\displaystyle\lim_{\varepsilon\to
0}\mathbb{E}\bigg{(}\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t\bigg{)}=0.$

###### Proof.

First we want to bound $\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t$
through a term that depends on
$\int^{T}_{0}(j_{t}-j_{t-2\varepsilon})^{2}\mathrm{d}t$ and apply this bound
inductively. For every $\varepsilon>0$, we have

$\displaystyle\int^{T}_{0}(j_{t}-j_{t-2\varepsilon})^{2}\mathrm{d}t=$
$\displaystyle\leavevmode\nobreak\
\int^{T}_{0}(j_{t}-j_{t-\varepsilon}+j_{t-\varepsilon}-j_{t-2\varepsilon})^{2}\mathrm{d}t$
$\displaystyle\geq$ $\displaystyle\leavevmode\nobreak\
\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t+\int_{0}^{T-\varepsilon}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
2\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t-\int_{T-\varepsilon}^{T}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t$

as well as

$\displaystyle\int_{T-\varepsilon}^{T}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t\leq\int_{T-\varepsilon}^{T}(j_{T}-j_{T-2\varepsilon})^{2}\mathrm{d}t\leq\varepsilon(j_{T}-j_{T-2\varepsilon})^{2}.$

As a result,

$\displaystyle\frac{1}{\varepsilon}\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t\leq\frac{1}{2\varepsilon}\int^{T}_{0}(j_{t}-j_{t-2\varepsilon})^{2}\mathrm{d}t+\frac{1}{2}(j_{T}-j_{T-2\varepsilon})^{2}.$

Hence, inductively, for all $\varepsilon\in(0;T)$ and $k\in\\{1,2,\ldots\\}$,
we have

$\displaystyle\frac{1}{\varepsilon}\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t\leq$
$\displaystyle\leavevmode\nobreak\
\frac{1}{2\varepsilon}\int^{T}_{0}(j_{t}-j_{t-2\varepsilon})^{2}\mathrm{d}t+\frac{1}{2}(j_{T}-j_{T-2\varepsilon})^{2}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{4\varepsilon}\int^{T}_{0}(j_{t}-j_{t-4\varepsilon})^{2}\mathrm{d}t+\frac{1}{4}(j_{T}-j_{T-4\varepsilon})^{2}+\frac{1}{2}(j_{T}-j_{T-2\varepsilon})^{2}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\ \ldots$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{2^{k}\varepsilon}\int^{T}_{0}(j_{t}-j_{t-2^{k}\varepsilon})^{2}\mathrm{d}t+\sum_{l=1}^{k}\frac{1}{2^{l}}(j_{T})^{2}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{2^{k}\varepsilon}\int^{T}_{0}(j_{t}-j_{t-2^{k}\varepsilon})^{2}\mathrm{d}t+(j_{T})^{2}.$

Hence, for all $k\in\mathbb{N}$,

$\displaystyle\mathbb{E}\bigg{(}\int^{T}_{0}(j_{t}-j_{t-\varepsilon})^{2}\mathrm{d}t\bigg{)}\leq\bigg{(}\frac{T}{2^{k}}+\varepsilon\bigg{)}\mathbb{E}\big{(}(j_{T})^{2}\big{)}.$
(4.1)

This shows the desired result. ∎

We are now ready to prove the approximation of arbitrary portfolio processes
by absolutely continuous ones.

###### Proof of Lemma 2.12.

By the triangle inequality, for all $\beta,\nu,\varepsilon>0$,

$\displaystyle\begin{split}&\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta,\beta,\nu,\varepsilon}_{t}-X^{\theta}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}}\\\
\leq&\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int^{t}_{(T-\varepsilon)\wedge
t}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
+\sum_{*\in\\{+,-\\}}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int_{0}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}\big{(}j^{*}_{s}-j^{*}_{s-\varepsilon}\big{)}\mathrm{d}s-j^{*}_{t}\bigg{)}^{2}\mathrm{d}t\bigg{)}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
+\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\frac{1}{\varepsilon}(t-(T-\varepsilon))^{+}X^{\theta,\beta,\nu,\varepsilon}_{T-\varepsilon}\bigg{)}^{2}\mathrm{d}t\bigg{)}}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\
+\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s-V_{t}\bigg{)}^{2}\mathrm{d}t\bigg{)}}.\end{split}$
(4.2)

We analyse the four terms separately. This first term can be bounded by

$\displaystyle\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int^{t}_{(T-\varepsilon)\wedge
t}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}\leq\sqrt{\mathbb{E}\bigg{(}\int^{T}_{T-\varepsilon}\bigg{(}\int^{t}_{T-\varepsilon}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}\leq\frac{\varepsilon^{\frac{3}{2}}\beta}{\nu}\leavevmode\nobreak\
\to\leavevmode\nobreak\ 0\quad\mbox{for }\varepsilon\to 0.$ (4.3)

Regarding the second term, for all $\varepsilon<T/2$,

$\displaystyle\int_{0}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}\big{(}j^{*}_{s}-j^{*}_{s-\varepsilon}\big{)}\mathrm{d}s-j^{*}_{t}=-\int_{((T-\varepsilon)\wedge
t)-\varepsilon}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}(j^{*}_{t}-j^{*}_{s})\mathrm{d}s.$

Using the monotonicity of the jump processes it follows from Lemma 4.1 that

$\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int_{0}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}\big{(}j^{*}_{s}-j^{*}_{s-\varepsilon}\big{)}\mathrm{d}s-j^{*}_{t}\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int_{((T-\varepsilon)\wedge
t)-\varepsilon}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}(j^{*}_{t}-j^{*}_{s})\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{\varepsilon}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int_{((T-\varepsilon)\wedge
t)-\varepsilon}^{(T-\varepsilon)\wedge
t}\big{(}j^{*}_{t}-j^{*}_{s}\big{)}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int_{0}^{T-\varepsilon}\big{(}j^{*}_{t}-j^{*}_{t-\varepsilon}\big{)}^{2}\mathrm{d}t\bigg{)}+\mathbb{E}\bigg{(}\int_{T-\varepsilon}^{T}\big{(}j^{*}_{t}-j^{*}_{T-2\varepsilon}\big{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}j^{*}_{t}-j^{*}_{t-\varepsilon}\big{)}^{2}\mathrm{d}t\bigg{)}+\varepsilon\mathbb{E}\Big{(}\big{(}j^{*}_{T}\big{)}^{2}\Big{)}}\leavevmode\nobreak\
\to\leavevmode\nobreak\ 0\quad\mbox{ for }\varepsilon\to 0.$ (4.4)

For the third term, we conclude from the Itô isometry and the definition of
$\xi^{\theta,\beta,\nu,\varepsilon}$ that

$\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\frac{1}{\varepsilon}(t-(T-\varepsilon))^{+}X^{\theta,\beta,\nu,\varepsilon}_{T-\varepsilon}\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\sqrt{\frac{\varepsilon}{3}\mathbb{E}\bigg{(}\bigg{(}x_{0}+\int_{0}^{T-\varepsilon}\xi^{\theta,\beta,\nu,\varepsilon}_{s}\mathrm{d}s\bigg{)}^{2}\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
x_{0}\sqrt{\frac{\varepsilon}{3}}+\sum_{*\in\\{+,-\\}}\sqrt{\frac{\varepsilon}{3}\mathbb{E}\bigg{(}\bigg{(}\int_{0}^{T-\varepsilon}\frac{1}{\varepsilon}(j^{*}_{s}-j^{*}_{s-\varepsilon})\mathrm{d}s\bigg{)}^{2}\bigg{)}}+\sqrt{\frac{\varepsilon}{3}\mathbb{E}\bigg{(}\bigg{(}\int_{0}^{T-\varepsilon}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\bigg{)}}.$
(4.5)

Now we can bound

$\displaystyle\sqrt{\frac{\varepsilon}{3}\mathbb{E}\bigg{(}\bigg{(}\int_{0}^{T-\varepsilon}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\bigg{)}}\leq
T\frac{\beta}{\nu}\sqrt{\frac{\varepsilon}{3}}\to 0\quad\mbox{for
}\varepsilon\to 0.$

Moreover, in view of (4.4),

$\displaystyle\leavevmode\nobreak\
\sqrt{\frac{\varepsilon}{3}\mathbb{E}\bigg{(}\bigg{(}\int_{0}^{T-\varepsilon}\frac{1}{\varepsilon}(j^{*}_{s}-j^{*}_{s-\varepsilon})\mathrm{d}s\bigg{)}^{2}\bigg{)}}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\sqrt{\frac{1}{3}\mathbb{E}\bigg{(}\int_{T-\varepsilon}^{T}\bigg{(}\int_{0}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}(j^{*}_{s}-j^{*}_{s-\varepsilon})\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\sqrt{\frac{1}{3}\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int_{0}^{(T-\varepsilon)\wedge
t}\frac{1}{\varepsilon}(j^{*}_{s}-j^{*}_{s-\varepsilon})\mathrm{d}s-j^{*}_{t}\bigg{)}^{2}\mathrm{d}t\bigg{)}}+\sqrt{\frac{1}{3}\mathbb{E}\bigg{(}\int_{T-\varepsilon}^{T}(j^{*}_{t})^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{\sqrt{3}}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}j^{*}_{t}-j^{*}_{t-\varepsilon}\big{)}^{2}\mathrm{d}t\bigg{)}+\varepsilon\mathbb{E}\Big{(}\big{(}j^{*}_{T}\big{)}^{2}\Big{)}}+\sqrt{\frac{\varepsilon}{3}\mathbb{E}\Big{(}\big{(}j^{*}_{T}\big{)}^{2}\Big{)}}\leavevmode\nobreak\
\to\leavevmode\nobreak\ 0\quad\mbox{for }\varepsilon\to 0.$

It remains to consider the fourth term in (4.2). To this end, let

$M^{\beta,\nu}:=\bigg{\\{}\sup_{t\in[0;T]}\bigg{|}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s-V_{t}\bigg{|}\leq
3\beta\bigg{\\}}.$

Then, using (2.17) in the last step,

$\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\bigg{(}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s-V_{t}\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\mathbbm{1}_{M^{\beta,\nu}}\bigg{(}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s-V_{t}\bigg{)}^{2}\mathrm{d}t\bigg{)}}+\sqrt{\mathbb{E}\bigg{(}\mathbbm{1}_{\Omega\backslash
M^{\beta,\nu}}\int^{T}_{0}(V_{t})^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leavevmode\nobreak\
+\sqrt{\mathbb{E}\bigg{(}\mathbbm{1}_{\Omega\backslash
M^{\beta,\nu}}\int^{T}_{0}\bigg{(}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
3\beta\sqrt{T}+\sqrt{T\mathbb{E}\bigg{(}\mathbbm{1}_{\Omega\backslash
M^{\beta,\nu}}\max_{t\in[0;T]}|V_{t}|^{2}\bigg{)}}+\sqrt{T\mathbb{E}\bigg{(}\mathbbm{1}_{\Omega\backslash
M^{\beta,\nu}}\max_{t\in[0;T]}\bigg{(}\int^{t}_{0}\dot{V}^{\beta,\nu}_{s}\mathrm{d}s\bigg{)}^{2}\bigg{)}}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
3\beta\sqrt{T}+2\sqrt{T\mathbb{E}\bigg{(}\mathbbm{1}_{\Omega\backslash
M^{\beta,\nu}}\max_{t\in[0;T]}|V_{t}|^{2}\bigg{)}}.$

By (2.15), $\\{\max_{t\in[0;T]}|V_{t}|^{2}\\}$ is uniformly integrable. Hence
we can first choose $\beta>0$ small enough, then, according to Lemma 2.11,
choose $\nu>0$ small enough such that $\mathbb{P}(\Omega\backslash
M^{\beta,\nu})$ is sufficiently small and finally choose $\varepsilon>0$ small
enough in order to obtain the desired result. ∎

###  4.3 Proof of Lemma 2.13

We start with a technical lemma.

######  Lemma 4.2.

Let $(M,\mathcal{A},\mu)$ be a measure space and let
$u,v\in\mathcal{L}^{2}(M)$. Then

$\displaystyle\int_{M}|u^{2}-v^{2}|\mathrm{d}\mu\leq\int_{M}(u-v)^{2}\mathrm{d}\mu+2\sqrt{\int_{M}(u-v)^{2}\mathrm{d}\mu\int_{M}v^{2}\mathrm{d}\mu}.$

###### Proof.

Due to the Hölder inequality and the triangle inequality,

$\displaystyle\int_{M}|u^{2}-v^{2}|\mathrm{d}\mu=$
$\displaystyle\sqrt{\int_{M}(u-v)^{2}\mathrm{d}\mu}\sqrt{\int_{M}(u+v)^{2}\mathrm{d}\mu}$
$\displaystyle\leq$
$\displaystyle\sqrt{\int_{M}(u-v)^{2}\mathrm{d}\mu}\Bigg{(}\sqrt{\int_{M}(u-v)^{2}\mathrm{d}\mu}+\sqrt{\int_{M}(2v)^{2}\mathrm{d}\mu}\Bigg{)}.$

∎

The following technical lemma provides useful estimates for the impact
process.

######  Lemma 4.3.

Let $X\in\mathcal{L}_{\mathcal{P}}^{2}(\Omega\times[0;T];\mathbb{R})$. Then
the transient price impact process $Y^{X}$ given by (2.1) satisfies
$Y^{X}\in\mathcal{L}_{\mathcal{P}}^{2}(\Omega\times[0;T];\mathbb{R})$ and

$\displaystyle\sqrt{\mathbb{E}\Big{(}\big{(}Y^{X}_{T}\big{)}^{2}\Big{)}}$
$\displaystyle\leq\leavevmode\nobreak\
\gamma\Bigg{(}X_{0}+\sqrt{\mathbb{E}\Big{(}\big{(}X_{T}\big{)}^{2}\Big{)}}+\overline{\rho}\exp(T\overline{\rho})\sqrt{T\mathbb{E}\bigg{(}\int^{T}_{0}(X_{s})^{2}\mathrm{d}s\bigg{)}}\Bigg{)},$
(4.6)
$\displaystyle\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}(Y^{X}_{t})^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq\leavevmode\nobreak\ \gamma
X_{0}\sqrt{T}+\gamma\big{(}1+T\overline{\rho}\exp(T\overline{\rho})\big{)}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}(X_{t})^{2}\mathrm{d}t\bigg{)}}.$
(4.7)

If $X_{0}=0$, then additionally, we have for all $s,t\in[0;T]$ with $s<t$,

$\displaystyle\sqrt{\mathbb{E}\bigg{(}\int_{s}^{t}\big{(}Y^{X}_{u}\big{)}^{2}\mathrm{d}u\bigg{)}}\leq$
$\displaystyle\leavevmode\nobreak\
\gamma\overline{\rho}\exp\big{(}T\overline{\rho}\big{)}\sqrt{(t-s)T\mathbb{E}\bigg{(}\int_{0}^{s}\big{(}X_{u}\big{)}^{2}\mathrm{d}u\bigg{)}\bigg{)}}$
$\displaystyle\leavevmode\nobreak\
+\gamma\Big{(}1+\overline{\rho}\exp\big{(}T\overline{\rho}\big{)}\sqrt{(t-s)T}\Big{)}\sqrt{\mathbb{E}\bigg{(}\int_{s}^{t}\big{(}X_{u}\big{)}^{2}\mathrm{d}u\bigg{)}}.$
(4.8)

###### Proof.

Inequality (4.6) follows from the explicit formula

$\displaystyle Y^{X}_{t}=\gamma
X_{t}-\gamma\exp\bigg{(}-\int^{t}_{0}\rho_{u}\mathrm{d}u\bigg{)}\bigg{(}X_{0}+\int^{t}_{0}\rho_{s}X_{s}\exp\bigg{(}\int_{0}^{s}\rho_{u}\mathrm{d}u\bigg{)}\mathrm{d}s\bigg{)}$
(4.9)

and the triangle inequality.

Moreover,

$\displaystyle\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}(Y^{X}_{t})^{2}\mathrm{d}t\bigg{)}}\leq\gamma\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}(X_{t})^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leavevmode\nobreak\
+\gamma\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\exp\bigg{(}-2\int^{t}_{0}\rho_{u}\mathrm{d}u\bigg{)}\bigg{(}X_{0}+\int^{t}_{0}\rho_{s}X_{s}\exp\bigg{(}\int_{0}^{s}\rho_{u}\mathrm{d}u\bigg{)}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
(4.10)

Substituting the inequality

$\displaystyle\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\exp\bigg{(}-2\int^{t}_{0}\rho_{u}\mathrm{d}u\bigg{)}\bigg{(}X_{0}+\int^{t}_{0}\rho_{s}X_{s}\exp\bigg{(}\int_{0}^{s}\rho_{u}\mathrm{d}u\bigg{)}\mathrm{d}s\bigg{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\mathbb{E}\big{(}T(X_{0})^{2}\big{)}}+\overline{\rho}\exp(T\overline{\rho})\sqrt{T\mathbb{E}\bigg{(}\bigg{(}\int^{T}_{0}|X_{s}|\mathrm{d}s\bigg{)}^{2}\bigg{)}}$
$\displaystyle\leq$ $\displaystyle
X_{0}\sqrt{T}+T\overline{\rho}\exp(T\overline{\rho})\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}(X_{s})^{2}\mathrm{d}s\bigg{)}}$
(4.11)

into (4.10) yields (4.7). To prove (4.8), let $X_{0}=0$. Then, due to (4.9),

$\displaystyle|Y^{X}_{u}-\gamma
X_{u}|\leq\gamma\overline{\rho}\exp\big{(}T\overline{\rho}\big{)}\int_{0}^{u}|X_{r}|\mathrm{d}r$

for all $u\in[0;T]$ and so

$\displaystyle\mathbb{E}\bigg{(}\int_{s}^{t}(Y^{X}_{u}-\gamma
X_{u})^{2}\mathrm{d}u\bigg{)}\leq(t-s)T\gamma^{2}\overline{\rho}^{2}\exp\big{(}2T\overline{\rho}\big{)}\mathbb{E}\bigg{(}\int^{t}_{0}(X_{u})^{2}\mathrm{d}u\bigg{)}.$

Using the subadditivity of the square root, we now obtain (4.8) from

$\displaystyle\sqrt{\mathbb{E}\bigg{(}\int_{s}^{t}(Y^{X}_{u})^{2}\mathrm{d}u\bigg{)}}\leq\sqrt{\mathbb{E}\bigg{(}\int_{s}^{t}(Y^{X}_{u}-\gamma
X_{u})^{2}\mathrm{d}u\bigg{)}}+\sqrt{\mathbb{E}\bigg{(}\int_{s}^{t}(\gamma
X_{u})^{2}\mathrm{d}u\bigg{)}}$ $\displaystyle\leq$
$\displaystyle\gamma\overline{\rho}\exp\big{(}T\overline{\rho}\big{)}\sqrt{(t-s)T\mathbb{E}\bigg{(}\int_{0}^{s}(X_{u})^{2}\mathrm{d}u\bigg{)}\bigg{)}}+\gamma\overline{\rho}\exp\big{(}T\overline{\rho}\big{)}\sqrt{(t-s)T\mathbb{E}\bigg{(}\int_{s}^{t}(X_{u})^{2}\mathrm{d}u\bigg{)}\bigg{)}}$
$\displaystyle+\gamma\sqrt{\mathbb{E}\bigg{(}\int_{s}^{t}(X_{u})^{2}\mathrm{d}u\bigg{)}}.$

∎

We are now ready to prove our approximation result for the cost functional.

###### Proof of Lemma 2.13.

For $\theta=(j^{+},j^{-},V)\in\mathcal{A}^{0}$ and $\xi\in\mathcal{A}$,

$\displaystyle\big{|}J^{0}(\theta)-J^{0}\big{(}0,0,V^{\xi}\big{)}\big{|}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\bigg{|}\mathbb{E}\bigg{(}\frac{\gamma}{2}(j^{+}_{0}-j^{-}_{0})^{2}+\int_{(0;T]}\frac{1}{2}(Y^{\theta}_{t-}+Y^{\theta}_{t})\mathrm{d}X^{\theta}_{t}-\int^{T}_{0}Y^{\xi}_{t}\mathrm{d}X^{\xi}_{t}+\frac{\gamma}{2}[V]_{T}\bigg{)}\bigg{|}$
$\displaystyle\leavevmode\nobreak\
+\frac{\overline{\lambda}}{2}\mathbb{E}\bigg{(}\int^{T}_{0}\Big{|}\big{(}X^{\theta}_{t}\big{)}^{2}-\big{(}X^{\xi}_{t}\big{)}^{2}\Big{|}\mathrm{d}t\bigg{)}.$

Due to Lemma 4.2 the last term can be estimated as

$\displaystyle\mathbb{E}\bigg{(}\int^{T}_{0}\Big{|}\big{(}X^{\theta}_{t}\big{)}^{2}-\big{(}X^{\xi}_{t}\big{)}^{2}\Big{|}\mathrm{d}t\bigg{)}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}+2\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}}$

Moreover, using first Lemma 3.10, and then Lemma 4.3 and Lemma 4.2, we obtain

$\displaystyle\leavevmode\nobreak\
\bigg{|}\mathbb{E}\bigg{(}\frac{\gamma}{2}(j^{+}_{0}-j^{-}_{0})^{2}+\int_{(0;T]}\frac{1}{2}(Y^{\theta}_{t-}+Y^{\theta}_{t})\mathrm{d}X^{\theta}_{t}-\int^{T}_{0}Y^{\xi}_{t}\mathrm{d}X^{\xi}_{t}+\frac{\gamma}{2}[V]_{T}\bigg{)}\bigg{|}$
$\displaystyle=$ $\displaystyle\leavevmode\nobreak\
\bigg{|}\mathbb{E}\bigg{(}\frac{\gamma}{2}(j^{+}_{0}-j^{-}_{0})^{2}+\frac{1}{2\gamma}\big{(}Y^{\theta}_{T}\big{)}^{2}-\frac{\gamma}{2}(j^{+}_{0}-j^{-}_{0})^{2}-\frac{\gamma}{2}[V]_{T}+\frac{1}{\gamma}\int^{T}_{0}\rho_{t}\big{(}Y^{\theta}_{t}\big{)}^{2}\mathrm{d}t$
$\displaystyle\leavevmode\nobreak\
-\frac{1}{2\gamma}\big{(}Y^{\xi}_{T}\big{)}^{2}-\frac{1}{\gamma}\int^{T}_{0}\rho_{t}\big{(}Y^{\xi}_{t}\big{)}^{2}\mathrm{d}t+\frac{\gamma}{2}[V]_{T}\bigg{)}\bigg{|}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\frac{1}{2\gamma}\mathbb{E}\bigg{(}\Big{|}\big{(}Y^{\theta}_{T}\big{)}^{2}-\big{(}Y^{\xi}_{T}\big{)}^{2}\Big{|}\bigg{)}+\frac{\overline{\rho}}{\gamma}\mathbb{E}\bigg{(}\int^{T}_{0}\Big{|}\big{(}Y^{\theta}_{t}\big{)}^{2}-\big{(}Y^{\xi}_{t}\big{)}^{2}\Big{|}\mathrm{d}t\bigg{)}$
$\displaystyle\leq$ $\displaystyle\leavevmode\nobreak\
\bigg{(}\frac{1}{2}T\gamma\overline{\rho}^{2}e^{2T\overline{\rho}}+\overline{\rho}\gamma\big{(}1+T\overline{\rho}e^{T\overline{\rho}}\big{)}^{2}\bigg{)}\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}$
$\displaystyle\leavevmode\nobreak\
+\sqrt{T}\gamma\overline{\rho}e^{T\overline{\rho}}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}}\Bigg{(}x_{0}+\overline{\rho}e^{T\overline{\rho}}\sqrt{T\mathbb{E}\bigg{(}\int^{T}_{0}(X^{\theta}_{t})^{2}\mathrm{d}t\bigg{)}}\Bigg{)}$
$\displaystyle\leavevmode\nobreak\
+2\overline{\rho}\big{(}1+T\overline{\rho}e^{T\overline{\rho}}\big{)}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}\big{(}X^{\theta}_{t}-X^{\xi}_{t}\big{)}^{2}\mathrm{d}t\bigg{)}}$
$\displaystyle\quad\Bigg{(}\gamma
x_{0}\sqrt{T}+\gamma\big{(}1+T\overline{\rho}e^{T\overline{\rho}}\big{)}\sqrt{\mathbb{E}\bigg{(}\int^{T}_{0}(X^{\theta}_{t})^{2}\mathrm{d}t\bigg{)}}\Bigg{)}.$

The assertion now follows from the fact that
$\mathbb{E}(\int^{T}_{0}(X^{\theta}_{t})^{2}\mathrm{d}t)<\infty$. ∎

##  Appendix A Regularity properties of Itô processes and BSDEs

In this appendix, we introduce some regularity properties of stochastic
processes, which we later use to prove various convergence results for
stochastic processes.

We consider a continuous, adapted, $\mathbb{R}^{d}$-valued stochastic process
$Y=\\{Y_{t}\\}_{t\in I}$ on some interval $I$ and introduce the following
continuity conditions.

######  Condition (C.1).

For all $\varepsilon\in(0;\infty)$, there exists $\delta\in(0;\infty)$ such
that, for all $s\in I$, all $\mathcal{F}_{s}$-measurable and integrable
$V\colon\Omega\to\mathbb{R}$ and all stopping times
$\tau\colon\Omega\to[s;s+\delta]\cap I$,

$\big{\|}\mathbb{E}\big{(}V(Y_{\tau}-Y_{s})\big{)}\big{\|}_{\infty}\leq\varepsilon\mathbb{E}\big{(}|V|\big{)}.$

######  Condition (C.2).

For all $\varepsilon\in(0;\infty)$, there exists $\delta\in(0;\infty)$ such
that, for all $s\in I$,

$\mathbb{E}\bigg{(}\sup_{t\in[s;s+\delta]\cap
I}\|Y_{t}-Y_{s}\|_{\infty}\,\Big{|}\,\mathcal{F}_{s}\bigg{)}\leq\varepsilon.$

######  Definition A.1.

A family of stochastic processes is said to uniformly satisfy Condition C.1 or
C.2 on $I$ if all processes satisfy the respective property and $\delta$ can
be chosen uniformly for all processes.

In what follows, we list some auxiliary results that will be useful later.

######  Lemma A.2.

An $\mathbb{R}^{d}$-valued stochastic process
$Y=\\{(Y^{1}_{t},\ldots,Y^{d}_{t})\\}_{t\in I}$ satisfies Condition C.1
(Condition C.2) if and only if all components $Y^{i}$ satisfy Condition C.1
(Condition C.2).

######  Lemma A.3.

Condition C.2 implies Condition C.1.888If we want to bound a term of the form
$\|\mathbb{E}(\mathbbm{1}_{N}(Y_{\tau}-Y_{s}))\|_{\infty}$ for some
$N\subseteq M\in\mathcal{F}_{s}$ with $N\not\in\mathcal{F}_{s}$, then
Condition C.1 is not enough; in this case, we need the stronger Condition C.2.

###### Proof.

Due to Lemma A.2, it is enough to consider a real-valued $Y$. The following
calculation shows that $Y$ satisfies Condition C.1 under the assumption that
it satisfies Condition C.2:

$\displaystyle\Big{|}\mathbb{E}\big{(}V(Y_{\tau}-Y_{s})\big{)}\Big{|}=\Big{|}\mathbb{E}\big{(}V\cdot\mathbb{E}(Y_{\tau}-Y_{s}\,|\,\mathcal{F}_{s})\big{)}\Big{|}\leq\mathbb{E}\big{(}|V|\cdot\mathbb{E}(|Y_{\tau}-Y_{s}|_{\infty}\,|\,\mathcal{F}_{s})\big{)}\leq\mathbb{E}(|V|)\varepsilon.$

∎

######  Lemma A.4.

Let $X=\\{X_{t}\\}_{t\in I}$ and $Y=\\{Y_{t}\\}_{t\in I}$ be continuous,
adapted, real-valued processes, which satisfy Condition C.1 (Condition C.2).
Then $X+Y$ satisfies Condition C.1 (Condition C.2).

######  Theorem A.5.

Let $X=\\{X_{t}\\}_{t\in I}$ and $Y=\\{Y_{t}\\}_{t\in I}$ be continuous,
adapted, essentially bounded, real-valued processes.

  * •

If $X$ satisfies Condition C.2 and $Y$ satisfies Condition C.1, then $X\cdot
Y$ satisfies Condition C.1.

  * •

If $X$ and $Y$ both satisfy Condition C.2, then $X\cdot Y$ also satisfies
Condition C.2.

###### Proof.

The second part of the statement can be proven straightforward using the
decomposition

$|X_{t}Y_{t}-X_{s}Y_{s}|\leq|X_{t}||Y_{t}-Y_{s}|+|Y_{s}||X_{t}-X_{s}|.$

The first part of the statement follows from

$\displaystyle\big{|}\mathbb{E}\big{(}V(X_{\tau}Y_{\tau}-X_{s}Y_{s})\big{)}\big{|}\leq\big{|}\mathbb{E}\big{(}VY_{\tau}(X_{\tau}-X_{s})\big{)}\big{|}+\big{|}\mathbb{E}\big{(}VX_{s}(Y_{\tau}-Y_{s})\big{)}\big{|}.$

∎

Next, we prove some properties of the concave envelope of the modulus of
continuity of a uniformly continuous function. We use these results to show
that a uniformly continuous function of an Itô process with bounded
coefficients satisfies Condition C.2.

######  Lemma A.6.

Let $(X,\|\cdot\|_{X})$ be a normed space with a non-empty and convex subset
$D$, let $(Y,d_{Y})$ be a metric space, let $f\colon D\to Y$ and, for every
$t\in[0;\infty)$ let

$\displaystyle\tilde{\omega}_{f}(t):=$ $\displaystyle\sup_{x_{1},x_{2}\in
D,\|x_{1}-x_{2}\|_{X}\leq t}d_{Y}(f(x_{1}),f(x_{2})),$
$\displaystyle\omega_{f}(t):=$
$\displaystyle\inf\\{\psi(t)\,|\,\psi\colon[0;\infty)\to[0;\infty)\,\text{concave
and}\,\,\psi\geq\tilde{\omega}_{f}\\}.$

Then $\tilde{\omega}_{f}\colon[0;\infty)\to[0;\infty]$, $\tilde{\omega}_{f}$
is non-decreasing, $\tilde{\omega}_{f}(0)=0$, and for all $s,t\in[0;\infty)$
with $s>0$,

$\displaystyle\tilde{\omega}_{f}(t)\leq\frac{t+s}{s}\tilde{\omega}_{f}(s).$
(A.1)

If $\tilde{\omega}_{f}$ is finite, then $\omega_{f}$ takes values in
$[0;\infty)$, is concave, non-decreasing and, for all $x_{1},x_{2}\in D$,

$d_{Y}(f(x_{1}),f(x_{2}))\leq\omega_{f}(\|x_{1}-x_{2}\|_{X}).$ (A.2)

If $f$ is uniformly continuous, then $\tilde{\omega}_{f}$ is finite,
$\tilde{\omega}_{f}$ and $\omega_{f}$ are continuous in $0$ and
$\omega_{f}(0)=0$.

###### Proof.

By definition, $\tilde{\omega}_{f}$ is non-decreasing and
$\tilde{\omega}_{f}(0)=0$. To prove (A.1), let $s,t\in[0;\infty)$ with $s>0$
and $x_{1},x_{2}\in D$ with $\|x_{1}-x_{2}\|_{X}\leq t$. Let $N:=\lceil
t/s\rceil$ and let

$\displaystyle x^{k}:=\frac{k}{N}x_{1}+\frac{N-k}{N}x_{2}\quad(k=0,\ldots,N).$

Then $x^{k}\in D$ because $D$ is convex and
$\|x^{k}-x^{k+1}\|_{X}=\frac{1}{N}\big{\|}x_{1}-x_{2}\big{\|}_{X}\leq s.$
Hence

$\displaystyle
d_{Y}(f(x_{1}),f(x_{2}))\leq\sum_{k=0}^{N-1}d_{Y}(f(x^{k}),f(x^{k+1}))\leq
N\max_{k}\tilde{\omega}_{f}\big{(}\|x^{k}-x^{k+1}\|_{X}\big{)}\leq(t+s)\frac{\tilde{\omega}_{f}(s)}{s}.$

Having established (A.1), we know that, if $\tilde{\omega}_{f}$ is finite,
then it is bounded above by an affine (hence concave) function. In particular,
the concave envelope is well-defined and satisfies (A.2).

Concavity and monotonicity of $\omega_{f}$ are clear.

Uniform continuity of $f$ implies that $\tilde{\omega}_{f}$ is finite and
continuous in $0$. In order to see $\omega_{f}(0)=0$ and continuity in $0$,
let $\varepsilon>0$ be arbitrary and let $\delta>0$ be such that
$\tilde{\omega}_{f}(\delta)\leq\varepsilon/2$. Due to (A.1),
$\tilde{\omega}_{f}$ and hence $\omega_{f}$ is bounded above by the affine and
hence concave function $t\mapsto((t+\delta)/\delta)\cdot(\varepsilon/2)$. In
particular, for all $t\leq\delta$, we obtain $\omega_{f}(t)\leq\varepsilon$. ∎

######  Lemma A.7.

Let $X=\\{X_{t}\\}_{t\in I}$ be a continuous, adapted, $\mathbb{R}^{d}$-valued
process which satisfies Condition C.2 and let
$f\colon[0;T]\times\mathbb{R}^{d}\to\mathbb{R}$ be uniformly continuous. Then
$\\{f(t,X_{t})\\}_{t\in[0;T]}$ also satisfies Condition C.2.

###### Proof.

There exists a function $\omega_{f}$ with the properties listed by Lemma A.6
(with the $\|\cdot\|_{\infty}$ norm on $[0;T]\times\mathbb{R}^{d}$). Now let
$\delta>0$ be arbitrary. Using (A.2) and the concavity and monotonicity of
$\omega_{f}$, we obtain that

$\displaystyle\mathbb{E}\bigg{(}\max_{s\leq t\leq
T\wedge(s+\delta)}|f(t,X_{t})-f(s,X_{s})|\bigg{|}\mathcal{F}_{s}\bigg{)}$
$\displaystyle\leq$ $\displaystyle\mathbb{E}\bigg{(}\max_{s\leq t\leq
T\wedge(s+\delta)}\omega_{f}\big{(}|t-s|\vee\|X_{t}-X_{s}\|_{\infty}\big{)}\bigg{|}\mathcal{F}_{s}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\bigg{(}\omega_{f}\bigg{(}\delta\vee\max_{s\leq t\leq
T\wedge(s+\delta)}\|X_{t}-X_{s}\|_{\infty}\bigg{)}\bigg{|}\mathcal{F}_{s}\bigg{)}$
$\displaystyle\leq$
$\displaystyle\omega_{f}\bigg{(}\delta+\mathbb{E}\bigg{(}\max_{s\leq t\leq
T\wedge(s+\delta)}\|X_{t}-X_{s}\|_{\infty}\bigg{|}\mathcal{F}_{s}\bigg{)}\bigg{)}.$

Since $\omega_{f}$ is continuous in $0$ and since $X$ satisfies Condition C.2,
we can choose $\delta>0$ small enough such that this term is not greater than
$\varepsilon$. ∎

######  Lemma A.8.

Let
$\\{\tilde{\sigma}_{t}\\}_{t\in[0;T]}\in\mathcal{L}^{\infty}_{\mathcal{P}}(\Omega\times[0;T];\mathbb{R}^{m})$.
Then $\\{\int^{t}_{0}\tilde{\sigma}_{u}\mathrm{d}W_{u}\\}_{t\in[0;T]}$
satisfies Condition C.2.

###### Proof.

Let $\overline{\sigma}>0$ be a component-wise bound on $\tilde{\sigma}$. Using
the Jensen inequality, a variation of Doob’s maximal inequality for
conditional expectations999In order to prove that the conditional expectation
w.r.t. $\mathcal{F}_{s}$ of the running maximum of a submartingale is bounded,
we apply the classical Doob’s maximal inequality concerning the conditional
measures w.r.t. all sets $A\in\mathcal{F}_{s}$ with positive probability. and
the Itô isometry for conditional expectations, we obtain

$\displaystyle\mathbb{E}\bigg{(}\max_{s\leq t\leq
T\wedge(s+\delta)}\bigg{|}\int^{t}_{s}\tilde{\sigma}_{u}\mathrm{d}W_{u}\bigg{|}\,\,\bigg{|}\mathcal{F}_{s}\bigg{)}\leq$
$\displaystyle\sqrt{\mathbb{E}\bigg{(}\max_{s\leq t\leq
T\wedge(s+\delta)}\bigg{|}\int^{t}_{s}\tilde{\sigma}_{u}\mathrm{d}W_{u}\bigg{|}^{2}\,\,\bigg{|}\mathcal{F}_{s}\bigg{)}}$
$\displaystyle\leq$ $\displaystyle
2\sqrt{\mathbb{E}\bigg{(}\bigg{|}\int_{s}^{T\wedge(s+\delta)}\tilde{\sigma}_{u}\mathrm{d}W_{u}\bigg{|}^{2}\,\,\bigg{|}\mathcal{F}_{s}\bigg{)}}$
$\displaystyle\leq$ $\displaystyle
2\sqrt{m\mathbb{E}\bigg{(}\int_{s}^{T\wedge(s+\delta)}m\overline{\sigma}^{2}\mathrm{d}u\,\,\bigg{|}\mathcal{F}_{s}\bigg{)}}\leq
2m\overline{\sigma}\sqrt{\delta}.$

∎

######  Remark A.9.

Obviously, we have the same result for a drift part of an Itô process, i.e. if
$\tilde{\mu}\in\mathcal{L}^{\infty}_{\textup{Prog}}(\Omega\times[0;T];\mathbb{R}^{m})$,
then $\\{\int^{t}_{0}\tilde{\mu}_{u}\mathrm{d}u\\}_{t\in[0;T]}$ satisfies
Condition C.2. However, we cannot expect even the weaker Condition C.1 under
weaker intregrability assumptions, as the following example shows. Let $T=2$,
$\tilde{\mu}_{u}:=\mathbbm{1}_{[1;2]}(u)\cdot|W_{1}|$, $\varepsilon:=1$, let
$\delta\in(0;1)$ be arbitrary and let $s:=1$, $\tau:=1+\delta$ and
$V:=\mathbbm{1}_{\\{\delta|W_{1}|>1\\}}$. Then

$\displaystyle\frac{|\mathbb{E}(V\int^{1+\delta}_{1}\tilde{\mu}_{u}\mathrm{d}u)|}{\mathbb{E}(|V|)}=\frac{\mathbb{E}(\mathbbm{1}_{\\{\delta|W_{1}|>1\\}}\cdot\delta|W_{1}|)|}{\mathbb{E}(\mathbbm{1}_{\\{\delta|W_{1}|\leq
1\\}})}>1.$

Finally, we prove that the strong Condition C.2 holds for a certain class of
BSDEs driven by forward SDEs. Specifically, we prove that the solution to the
BSDE can be expressed as a uniformly continuous function of the forward
process and then we apply Lemma A.7. The representation of the solution in
terms of a continuous function has been proven in [12] already.

For all $(t,x)\in[0;T]\times\mathbb{R}^{n}$, we consider the following SDE on
$[t;T]$,

$\displaystyle\mathrm{d}X^{t,x}_{s}=\tilde{\mu}\big{(}s,X^{t,x}_{s}\big{)}\mathrm{d}s+\tilde{\sigma}\big{(}s,X^{t,x}_{s}\big{)}\mathrm{d}W_{s},\quad
X^{t,x}_{t}=x,$

where
$\tilde{\mu},\tilde{\sigma}\colon[0;T]\times\mathbb{R}^{n}\to\mathbb{R}^{n}\times\mathbb{R}^{n\times
m}$ are measurable and bounded and satisfy the Lipschitz condition

$\|(\tilde{\mu},\tilde{\sigma})(t,x_{1})-(\tilde{\mu},\tilde{\sigma})(t,x_{2})\|_{\infty}\leq
L\|x_{1}-x_{2}\|_{\infty}.$

By the previous results, we obtain that $X^{t,x}$ satisfies Condition C.2.

Standard computations show that there exists a constant $C\in(0;\infty)$ such
that, for all $t\in[0;T]$ and $x_{1},x_{2}\in\mathbb{R}^{n}$,

$\displaystyle\mathbb{E}\bigg{(}\sup_{s\in[t;T]}\|X^{t,x_{1}}_{s}-X^{t,x_{2}}_{s}\|_{\infty}^{2}\bigg{)}\leq
C\|x_{1}-x_{2}\|_{\infty}^{2}.$ (A.3)

######  Theorem A.10.

Let $\Psi\colon\mathbb{R}^{n}\to\mathbb{R}$ be bounded,
$\psi\colon[0;T]\times\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ be
continuous and bounded and let both function satisfy the Lipschitz condition

$|\Psi(x_{1})-\Psi(x_{2})|+|\psi(t,x_{1},y_{1})-\psi(t,x_{2},y_{2})|\leq
L\big{(}\|x_{1}-x_{2}\|_{\infty}+|y_{1}-y_{2}|\big{)}.$

For $(t,x)\in[0;T]\times\mathbb{R}^{n}$, let
$(Y^{t,x},Z^{t,x})\in\mathcal{L}_{\mathcal{P}}^{2}(\Omega\times[0;T];\mathbb{R}\times\mathbb{R}^{m})$
be the unique solution to the BSDE on $[t;T]$,

$\displaystyle-\mathrm{d}Y^{t,x}_{s}=\psi\big{(}s,X^{t,x}_{s},Y^{t,x}_{s}\big{)}\mathrm{d}s-Z^{t,x}_{s}\mathrm{d}W_{s},\quad
Y^{t,x}_{T}=\Psi\big{(}X^{t,x}_{T}\big{)}$ (A.4)

driven by the forward process $X^{t,x}$, which exists due to Theorem 2.1 in
[12]. Then there exists a uniformly continuous function
$h\colon[0;T]\times\mathbb{R}^{n}\to\mathbb{R}$, which does not depend on
$(t,x)$, such that

$\displaystyle\mathbb{P}\Big{[}h\big{(}s,X^{t,x}_{s}\big{)}=Y^{t,x}_{s}\mbox{
for all }s\in[t;T]\Big{]}=1.$ (A.5)

In particular, the process $Y^{t,x}$ satisfies Condition C.2 on $[t;T]$.

###### Proof.

By Theorem 3.4 in [12] and the remark following it, there exists a function
$h\colon[0;T]\times\mathbb{R}^{n}\to\mathbb{R}$ that is $1/2$-Hölder
continuous in $t$ and locally Lipschitz continuous in $x$ such that (A.5)
holds. It therefore remains to prove the uniform continuity of $h$. Since

$\displaystyle
h(t,x)=\mathbb{E}\bigg{(}\Psi\big{(}X^{t,x}_{T}\big{)}+\int_{t}^{T}\psi\Big{(}u,X^{t,x}_{u},h\big{(}u,X^{t,x}_{u}\big{)}\Big{)}\mathrm{d}u\bigg{)},$
(A.6)

we see that $h$ is bounded, say $|h|\leq\overline{h}$. Then

$\displaystyle\tilde{\omega}_{h}(s,\delta):=\sup_{\|x_{1}-x_{2}\|_{\infty}\leq\delta}|h(s,x_{1})-h(s,x_{2})|$

is finite, and Lemma A.6 allows us to define its concave envelope
$\omega_{h}(s,\cdot)\colon[0;\infty)\to[0;\infty)$.

Using the Lipschitz continuity of $\psi$, we obtain, for all
$x_{1},x_{2}\in\mathbb{R}^{n}$ and $t\in[0;T]$ that

$\begin{split}&\big{|}h(t,x_{1})-h(t,x_{2})\big{|}\\\
&\leq\leavevmode\nobreak\
L\mathbb{E}(\|X^{t,x_{1}}_{T}-X^{t,x_{2}}_{T}\|_{\infty})\\\
&\qquad+L\int_{t}^{T}\bigg{(}\mathbb{E}\Big{(}\big{\|}X^{t,x_{1}}_{u}-X^{t,x_{2}}_{u}\big{\|}_{\infty}\Big{)}+\mathbb{E}\Big{(}\tilde{\omega}_{h}\Big{(}u,\big{\|}X^{t,x_{1}}_{u}-X^{t,x_{2}}_{u}\big{\|}_{\infty}\Big{)}\Big{)}\bigg{)}\mathrm{d}u.\end{split}$
(A.7)

Using (A.3) along with Lemma A.6, if $\|x_{1}-x_{2}\|_{\infty}\leq\delta$,
then

$\displaystyle\mathbb{E}\Big{(}\big{\|}X^{t,x_{1}}_{T}-X^{t,x_{2}}_{T}\big{\|}_{\infty}\Big{)}$
$\displaystyle\leq C\delta$ (A.8)

and

$\displaystyle\mathbb{E}\Big{(}\tilde{\omega}_{h}\big{(}u,\big{\|}X^{t,x_{1}}_{u}-X^{t,x_{2}}_{u}\big{\|}_{\infty}\big{)}\Big{)}$
$\displaystyle\leq\omega_{h}\Big{(}u,\mathbb{E}\big{(}\big{\|}X^{t,x_{1}}_{u}-X^{t,x_{2}}_{u}\big{\|}_{\infty}\big{)}\Big{)}\leq\omega_{h}(u,C\delta),$
(A.9)

for some constant $C\in(0;\infty)$ that depends only on $\tilde{\mu}$,
$\tilde{\sigma}$ and $L$. In view of (A.1),

$\displaystyle\omega_{h}(u,C\delta)\leq(C+1)\tilde{\omega}_{h}(u,\delta).$
(A.10)

This shows that

$\displaystyle\tilde{\omega}_{h}(t,\delta)$
$\displaystyle\leq\leavevmode\nobreak\
LC\delta+L\int_{t}^{T}\big{(}C\delta+(C+1)\tilde{\omega}_{h}(u,\delta)\big{)}\mathrm{d}u$
$\displaystyle=\leavevmode\nobreak\
LC\delta(1+T-t)+L(C+1)\int_{t}^{T}\tilde{\omega}_{h}(u,\delta)\mathrm{d}u.$

Hence the Gronwall inequality implies that $h$ is uniformly Lipschitz
continuous in the second variable. Since $h$ is uniformly $1/2$-Hölder
continuous in the first variable, this proves the assertion. ∎

##  Appendix B Convergence results for SDEs and random ODEs

In this appendix, we establish three convergence results for stochastic
integral equations that are useful to establish the convergence of our
coefficient and state processes. In Subsection B.1, we consider sequences of
SDEs parametrized by $\eta\to 0$ with “positive feedback”, i.e. sequences of
equations that are driven away from their limits. In Subsection B.2, we
consider sequences of random ODE systems parametrized by $\eta\to 0$ with
“negative feedback”, which are driven towards their limit.

For equations with positive feedback, we require a priori information about
the terminal value and the increments of the coefficient processes to be
bounded independently of their past (see Condition C.1 and Condition C.2).
These bounds prevent the stochastic process from reaching its terminal value
if a large difference between the limit and the pre-limit occurs at some time
$s\in(0;T)$. This will imply a.s. uniform convergence. An almost sure
statement under negative feedback cannot be expected. Instead, we follow a
pathwise approach and establish uniform convergence in probability.

###  B.1 SDEs with positive feedback

####  B.1.1 An equation without scaling

Let us consider a family of real-valued, adapted, continuous stochastic
processes $b^{\eta}=\\{b^{\eta}_{t}\\}_{t\in(0;T)}$ that satisfy the integral
equation

$\displaystyle\mathrm{d}b^{\eta}_{t}=\Big{(}a\big{(}t,b^{\eta}_{t}\big{)}+q^{\eta}_{t}\Big{)}\mathrm{d}t+\overline{Z}^{\eta}_{t}\mathrm{d}W_{t}\quad\mbox{on
}(0;T)$ (B.1)

where $a\colon\Omega\times(0;T)\times\mathbb{R}\to\mathbb{R}$ is adapted and
continuous and $\\{q^{\eta}_{t}\\}_{t\in(0;T)}$ is an adapted, real-valued,
continuous stochastic process and

$\\{\overline{Z}^{\eta}_{t}\\}_{t\in(0;T)}\in\mathcal{L}_{\textup{Prog}}^{2}(\Omega\times(0;T-];\mathbb{R}^{m}).$

Our goal is to prove that the processes $\\{b^{\eta}_{t}\\}_{t\in(0;T)}$ with
terminal conditions $b^{\eta}_{T}=0$ converge to $0$ as $\eta\to 0$ uniformly
on $(0;T)$ if the process $q^{\eta}$ does and if the mapping $a$ is such that
$b^{\eta}$ is driven away from $0$. Intuitively, the last condition makes it
impossible for $b^{\eta}$ to return to $0$ at the terminal time once it has
left a neighbourhood of $0$. Specifically, we assume that the following
assumption is satisfied.

######  Assumption B.1.

  * i)

There exists a real constant $\overline{a}>0$ such that

$\sup_{t\in(0;T)}\sup_{\eta>0}\Big{|}a\big{(}t,b^{\eta}_{t}\big{)}\Big{|}\leq\overline{a}.$

  * ii)

For all $\varepsilon>0$, there exists a $\delta>0$ such that, for all
$\eta>0$,

$\mathbb{P}\bigg{[}\sup_{t\in[T-\delta;T)}\big{|}b^{\eta}_{t}\big{|}\leq\varepsilon\bigg{]}=1.$

  * iii)

There exists a function $\eta_{1}\colon(0;\infty)\to(0;\infty)$ such that, for
all $\varepsilon>0$ and $\eta\leq\eta_{1}(\varepsilon)$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{t\in(0;T-\varepsilon]}\big{|}q^{\eta}_{t}\big{|}\leq\varepsilon\bigg{]}=1.$

  * iv)

There exists an $\varepsilon_{1}>0$ such that

$\inf_{t\in(0;T)}\inf_{b\in[-\varepsilon_{1};\varepsilon_{1}]}b\cdot
a(t,b)\geq 0.$

######  Lemma B.2.

Under Assumption B.1, there exists a function
$\eta_{0}\colon(0;\infty)\to(0;\infty)$, which depends only on $\eta_{1}$ and
the mapping $\varepsilon\mapsto\delta$ introduced in Assumption B.1 ii), such
that, for all $\varepsilon>0$ and all $\eta\in(0;\eta_{0}(\varepsilon)]$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{s\in(0;T)}\big{|}b^{\eta}_{s}\big{|}\leq\varepsilon\bigg{]}=1.$

###### Proof.

Let $\varepsilon\in(0;\varepsilon_{1}]$. In a first step, we show that there
exists an $\eta_{0}(\varepsilon)>0$ such that

$\displaystyle\sigma^{\varepsilon,\eta}:=\inf\\{t\in(0;T)\,|\,\mathbb{P}[b^{\eta}_{s}<-\varepsilon]=0\text{
for all }s\in[t;T)\\}$

satisfies $\sigma^{\varepsilon,\eta}=0$ for all
$\eta\in(0;\eta_{0}(\varepsilon)]$. Since $b^{\eta}$ is continuous, this will
imply that, for all $\eta\in(0;\eta_{0}(\varepsilon)]$,

$\mathbb{P}\bigg{[}\inf_{s\in(0;T)}b^{\eta}_{s}\geq-\varepsilon\bigg{]}=1.$

The other direction of the statement can be proven analogously.

We now assume to the contrary that $0<\sigma^{\varepsilon,\eta}$ and prove
that this leads to a contradiction if $\eta$ is small enough. For all
$s\in(0;T)$ and $\eta\in(0;\infty)$, we can define the stopping time

$\displaystyle\tau^{s}:=\tau^{s,\varepsilon,\eta}:=\inf\\{u\in[s;T)\,|\,b^{\eta}_{u}=-\varepsilon/2\\}.$

By Assumption B.1 ii), there exists a small enough $\delta>0$ such that, for
all $s\in(0;T)$ and $\eta\in(0;\infty)$,

$\displaystyle\mathbb{P}\big{[}b^{\eta}_{s}<-\varepsilon,\tau^{s}>T-\delta\big{]}=0.$
(B.2)

Since $\tau^{s}$ is a stopping time, due to the square-integrability of
$\overline{Z}^{\eta}$, by the optional sampling theorem (cf. Theorem 1.3.22 in
[23]) and Assumption B.1 i), iv), (B.1) and (B.2), we obtain that, for all
$s\in(0;\sigma^{\varepsilon,\eta})$ and $\eta\in(0;\infty)$,

$\displaystyle\begin{split}\mathbb{P}[b^{\eta}_{s}<-\varepsilon]\frac{\varepsilon}{2}\leq&\leavevmode\nobreak\
\mathbb{E}\Big{(}\mathbbm{1}_{\\{b^{\eta}_{s}<-\varepsilon\\}}\big{(}b^{\eta}_{\tau^{s}}-b^{\eta}_{s}\big{)}\Big{)}\\\
=&\leavevmode\nobreak\
\mathbb{E}\Bigg{(}\mathbbm{1}_{\\{b^{\eta}_{s}<-\varepsilon\\}}\bigg{(}\int_{s}^{\tau^{s}\wedge\sigma^{\varepsilon,\eta}}a\big{(}t,b^{\eta}_{t}\big{)}\mathrm{d}t+\int_{\tau^{s}\wedge\sigma^{\varepsilon,\eta}}^{\tau^{s}}a\big{(}t,\underbrace{b^{\eta}_{t}}_{\in[-\varepsilon;-\varepsilon/2]\,\text{if}\,\sigma^{\varepsilon,\eta}<\tau^{s}}\big{)}\mathrm{d}t+\int_{s}^{\tau^{s}}q^{\eta}_{t}\mathrm{d}t\bigg{)}\Bigg{)}\\\
\leq&\leavevmode\nobreak\
\mathbb{P}\big{[}b^{\eta}_{s}<-\varepsilon\big{]}\bigg{(}\big{(}\sigma^{\varepsilon,\eta}-s\big{)}\overline{a}+T\sup_{t\in[0;T-\delta]}\big{|}q^{\eta}_{t}\big{|}\bigg{)}.\end{split}$
(B.3)

The assumption $0<\sigma^{\varepsilon,\eta}$ implies that for any $\alpha>0$,
we can find

$\sigma^{\varepsilon,\eta}-\alpha\leq s\leq\sigma^{\varepsilon,\eta}$

with $\mathbb{P}[b^{\eta}_{s}<-\varepsilon]>0$. However, due to Assumption B.1
ii) and iii), we can choose $\alpha\leq\varepsilon/(5\overline{a})$,
$\delta\leq\varepsilon/(5T)$ and $\eta_{0}(\varepsilon)<\eta_{1}(\delta)$ such
that (B.3) leads to a contradiction because
$\mathbb{P}[b^{\eta}_{s}<-\varepsilon]>0$. This implies
$0=\sigma^{\varepsilon,\eta}$. ∎

####  B.1.2 An equation with scaling

In this section, we establish an abstract convergence result for stochastic
processes $\\{P^{\eta}_{t}\\}_{t\in(0;T)}$ indexed by some parameter $\eta>0$
that satisfy the integral equation

$\displaystyle\mathrm{d}\big{(}\psi
P^{\eta}\big{)}_{t}=\sqrt{\eta^{-1}}\Big{(}a\big{(}P^{\eta}_{t},P^{0}_{t}\big{)}+q^{\eta}_{t}\Big{)}\mathrm{d}t+\mathrm{d}L^{\eta}_{t}+\overline{Z}^{\eta}_{t}\mathrm{d}W_{t}\quad\mbox{on
}(0;T)$ (B.4)

where $a:\mathbb{R}^{2}\to\mathbb{R}$ is a measurable mapping,
$\\{\psi_{t}\\}_{t\in(0;T)}$, $\\{P^{\eta}_{t}\\}_{t\in(0;T)}$,
$\\{P^{0}_{t}\\}_{t\in(0;T)}$, $\\{q^{\eta}_{t}\\}_{t\in(0;T)}$
$\\{L^{\eta}_{t}\\}_{t\in(0;T)}$ and are adapted, real-valued, continuous
stochastic processes and

$\\{\overline{Z}^{\eta}_{t}\\}_{t\in(0;T)}\in\mathcal{L}_{\textup{Prog}}^{2}(\Omega\times(0;T-];\mathbb{R}^{m}).$

Our goal is to prove that the processes $\\{P^{\eta}_{t}\\}_{t\in(0;T)}$
converge to $P^{0}$ as $\eta\to 0$ uniformly on compact subintervals of
$(0;T)$ if the mapping $a(\cdot,\cdot)$ is such that $P^{\eta}$ is driven away
from $P^{0}$ and if the boundary condition $\lim_{t\to
T}(P^{\eta}_{t}-P^{0}_{t})\geq 0$ holds. If the integral equation (B.4) holds
on the whole interval $(0;T]$, then it is enough to assume that
$P^{\eta}_{T}\geq P^{0}_{T}$. When applying the abstract convergence result to
the BSDEs (3.3) and (3.4), the former condition holds if $N=+\infty$ while the
latter holds if $N$ is finite.

######  Assumption B.3.

  * i)

There exist positive constants $\underline{\psi},\overline{\psi},\overline{P}$
such that $\underline{\psi}\leq\psi\leq\overline{\psi}$,
$|P^{0}|\leq\overline{P}$ and $P^{\eta}\geq-\overline{P}$. Moreover, there
exists a function $\eta_{1}\colon(0;\infty)\to(0;\infty)$ such that

$P^{\eta}|_{\Omega\times(0;T-\varepsilon]}\leq\overline{P}\quad\mbox{for
all}\quad\varepsilon>0\mbox{ and }\eta\leq\eta_{1}(\varepsilon).$

  * ii)

One of the following two “boundary conditions” holds:

    * a)

For all $\eta>0$, there exists a $T_{\eta}<T$ such that

$\mathbb{P}\bigg{[}\inf_{T_{\eta}\leq
t<T}\big{(}P^{\eta}_{t}-P^{0}_{t}\big{)}\geq 0\bigg{]}=1.$

    * b)

The integral equation (B.4) holds on the whole interval $(0;T]$.101010In
particular, we assume that $\psi$, $P^{\eta}$, $P^{0}$, $q^{\eta}$ and
$L^{\eta}$ are continuous on $(0;T]$ and
$\overline{Z}^{\eta}\in\mathcal{L}_{\textup{Prog}}^{2}(\Omega\times(0;T];\mathbb{R}^{m})$.
and $P^{\eta}_{T}\geq P^{0}_{T}$.

  * iii)

There exists a function $\eta_{2}\colon(0;\infty)\to(0;\infty)$ such that, for
all $\varepsilon>0$ and all $\eta\in(0;\eta_{2}(\varepsilon)]$,

$\displaystyle\mathbb{P}\bigg{[}\inf_{t\in(0;T-\varepsilon]}q^{\eta}_{t}\geq-\varepsilon,\sup_{t\in(0;T)}q^{\eta}_{t}\leq\varepsilon\bigg{]}=1.$

  * iv)

The function $a\colon\mathbb{R}^{2}\to\mathbb{R}$ is non-decreasing and
differentiable in the first variable and
$a(P^{0}_{t}(\omega),P^{0}_{t}(\omega))\equiv 0$. Moreover, there exist
$\overline{\varepsilon},\beta>0$ such that

$\frac{\partial}{\partial x}a(x,P^{0}_{t}(\omega))\geq\beta\quad\mbox{for all
}|x-P^{0}_{t}(\omega)|\leq\overline{\varepsilon}.$

  * v)

The processes $\psi$ and $P^{0}$ satisfy Condition C.2 and the processes
$L^{\eta}$ $(\eta>0)$ satisfy Condition C.1 uniformly in $\eta>0$.

######  Lemma B.4.

Let $p^{\eta}:=P^{\eta}-P^{0}$. Under Assumption B.3, there exists a function
$\eta_{0}\colon(0;\infty)\to(0;\infty)$, which depends only on $\psi$,
$\overline{P}$, $\eta_{1}$, $\eta_{2}$, $P^{0}$, $\overline{\varepsilon}$,
$\beta$ and the mapping $\varepsilon\mapsto\delta$ corresponding to the
uniform satisfaction of Condition C.1 by $\\{L^{\eta}\\}_{\eta}$, such that,
for all $\varepsilon>0$ and $\eta\in(0;\eta_{0}(\varepsilon)]$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{s\in(0;T-\varepsilon]}p^{\eta}_{s}\leq\varepsilon,\inf_{s\in(0;T)}p^{\eta}_{s}\geq-\varepsilon\bigg{]}=1.$

###### Proof.

Let us fix $\varepsilon\in(0;\overline{\varepsilon}]$. We first show that
there exists $\eta_{0}(\varepsilon)>0$ such that

$\sup_{\eta\leq\eta_{0}(\varepsilon)}\sup_{s\in(0;T-\varepsilon]}\mathbb{P}[p^{\eta}_{s}>\varepsilon]=0.$
(B.5)

Since $p^{\eta}$ is continuous, this proves that, for all
$\eta\leq\eta_{0}(\varepsilon)$,

$\mathbb{P}\bigg{[}\sup_{s\in(0;T-\varepsilon]}p^{\eta}_{s}\leq\varepsilon\bigg{]}=1.$

To this end, we fix $\delta\leq\varepsilon/2$ and $s\in(0;T-\varepsilon]$. For
all $\eta>0$, we can define the stopping time

$\displaystyle\tau^{\eta}:=(s+\delta)\wedge\inf\\{u\in[s;T)\,|\,|p^{\eta}_{u}|=\varepsilon/2\\}\leq
T-\delta.$

Due to the $L^{2}$-integrability of $\overline{Z}^{\eta}$ on $[s;T-\delta]$
and by the optional sampling theorem (cf. Theorem 1.3.22 in [23]), we obtain

$\begin{split}0=&\leavevmode\nobreak\
\mathbb{E}\bigg{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\\}}\bigg{(}\sqrt{\eta^{-1}}\int_{s}^{\tau^{\eta}}\Big{(}a\big{(}P^{\eta}_{u},P^{0}_{u}\big{)}+q^{\eta}_{u}\Big{)}\mathrm{d}u+L^{\eta}_{\tau^{\eta}}-L^{\eta}_{s}\\\
&\quad+\psi_{s}P^{0}_{s}-\psi_{\tau^{\eta}}P^{0}_{\tau^{\eta}}+\psi_{s}p^{\eta}_{s}-\psi_{\tau^{\eta}}p^{\eta}_{\tau^{\eta}}\bigg{)}\bigg{)}.\end{split}$
(B.6)

We now analyze the various terms in (B.6) separately to deduce that
$\mathbb{P}[p^{\eta}_{s}>\varepsilon]=0$ if

$\eta\leq\eta_{0}(\varepsilon):=\min\big{(}\eta_{1}(\varepsilon),\eta_{2}(\min(\delta,\beta\varepsilon/4))\big{)}.$

  * •

We first consider the term

$\mathbb{E}\bigg{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\\}}\sqrt{\eta^{-1}}\int_{s}^{\tau^{\eta}}\Big{(}a\big{(}P^{\eta}_{u},P^{0}_{u}\big{)}+q^{\eta}_{u}\Big{)}\mathrm{d}u\bigg{)}.$

On $\\{p^{\eta}_{s}>\varepsilon\\}$, we have that
$p^{\eta}_{u}\geq\varepsilon/2$ for all $u\in[s;\tau^{\eta}]$ and hence

$\displaystyle
a\big{(}P^{\eta}_{u},P^{0}_{u}\big{)}=a\big{(}P^{0}_{u}+p^{\eta}_{u},P^{0}_{u}\big{)}\geq
a\big{(}P^{0}_{u}+\varepsilon/2,P^{0}_{u}\big{)}\geq\frac{\beta\varepsilon}{2}.$

Together with our assumption on the process $q^{\eta}$ and our choice of
$\eta$, this implies that

$\begin{split}&\mathbb{E}\bigg{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\\}}\sqrt{\eta^{-1}}\int_{s}^{\tau^{\eta}}\Big{(}\underbrace{a\big{(}P^{\eta}_{u},P^{0}_{u}\big{)}}_{\geq\beta\varepsilon/2}+\underbrace{q^{\eta}_{u}}_{\geq-\beta\varepsilon/4}\Big{)}\mathrm{d}u\bigg{)}\\\
\geq&\mathbb{E}\bigg{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\\}}\sqrt{\eta^{-1}}(\tau^{\eta}-s)\frac{\beta\varepsilon}{4}\bigg{)}\\\
\geq&\delta\sqrt{\eta^{-1}}\frac{\beta\varepsilon}{4}\mathbb{P}\big{[}p^{\eta}_{s}>\varepsilon\text{
and }\tau^{\eta}=s+\delta\big{]}.\end{split}$ (B.7)

  * •

By Assumption B.3 v), Lemma A.3 and Theorem A.5, we can choose for any
$\varepsilon_{0}>0$ some $\delta>0$, which does not depend on $\eta$, such
that

$\Big{|}\mathbb{E}\Big{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\\}}(Y_{\tau^{\eta}}-Y_{s})\Big{)}\Big{|}\leq\varepsilon_{0}\mathbb{P}[p^{\eta}_{s}>\varepsilon],\quad\mbox{for
}Y=\psi P^{0},L^{\eta}.$ (B.8)

and

$\mathbb{E}\big{(}|\psi_{\tau^{\eta}}-\psi_{s}|\big{|}\mathcal{F}_{s}\big{)}\leq\varepsilon_{0}$
(B.9)

  * •

Using (B.8), along with the fact that
$P^{\eta}|_{\Omega\times(0;T-\varepsilon]}\leq\overline{P}$ due to Assumption
B.3 i), yields

$\begin{split}&\mathbb{E}\Big{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\\}}\big{(}\psi_{s}p^{\eta}_{s}-\psi_{\tau^{\eta}}p^{\eta}_{\tau^{\eta}}\big{)}\Big{)}\\\
\geq&\leavevmode\nobreak\
\mathbb{E}\Big{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\text{ and
}{\tau^{\eta}}<s+\delta\\}}\big{(}\psi_{s}\varepsilon/2+(\psi_{s}-\psi_{\tau^{\eta}})\varepsilon/2\big{)}\Big{)}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\mathbb{E}\Big{(}\mathbbm{1}_{\\{p^{\eta}_{s}>\varepsilon\text{ and
}{\tau^{\eta}}=s+\delta\\}}\big{(}\psi_{s}\varepsilon-\psi_{\tau^{\eta}}\underbrace{p^{\eta}_{\tau^{\eta}}}_{\leq
2\overline{P}}\big{)}\Big{)}\\\ \geq&\leavevmode\nobreak\
\frac{\varepsilon}{2}\underline{\psi}\mathbb{P}\big{[}p^{\eta}_{s}>\varepsilon\text{
and
}{\tau^{\eta}}<s+\delta\big{]}-\frac{\varepsilon\varepsilon_{0}}{2}\mathbb{P}\big{[}p^{\eta}_{s}>\varepsilon\big{]}\\\
&\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+\big{(}\varepsilon\underline{\psi}-2\overline{\psi}\overline{P}\big{)}\mathbb{P}\big{[}p^{\eta}_{s}>\varepsilon\text{
and }{\tau^{\eta}}=s+\delta\big{]}.\end{split}$ (B.10)

Altogether this yields

$\displaystyle 0\geq$
$\displaystyle\mathbb{P}\big{[}p^{\eta}_{s}>\varepsilon\text{ and
}{\tau^{\eta}}=s+\delta\big{]}\bigg{(}\delta\sqrt{\eta_{0}(\varepsilon)^{-1}}\frac{\beta\varepsilon}{4}-2\varepsilon_{0}-\frac{\varepsilon\varepsilon_{0}}{2}+\varepsilon\underline{\psi}-2\overline{\psi}\overline{P}\bigg{)}$
$\displaystyle+\mathbb{P}\big{[}p^{\eta}_{s}>\varepsilon\text{ and
}{\tau^{\eta}}<s+\delta\big{]}\bigg{(}-2\varepsilon_{0}+\frac{\varepsilon}{2}\underline{\psi}-\frac{\varepsilon\varepsilon_{0}}{2}\bigg{)}.$

Now, if we choose first $\varepsilon_{0}$ and then $\eta_{0}(\varepsilon)$
small enough, the coefficients that multiply the probabilities become
positive. Hence both probabilities must be equal to zero and so
$\mathbb{P}[p^{\eta}_{s}>\varepsilon]=0$ if $\eta\leq\eta_{0}(\varepsilon)$.
Analogously, we can prove that $\mathbb{P}[p^{\eta}_{s}<-\varepsilon]=0$ for
all $s\in(0;T)$. The main difference is that ${\tau^{\eta}}<T-\delta$ does not
hold on the set $\\{p^{\eta}_{s}<-\varepsilon\\}$. Instead, we only obtain
$\tau^{\eta}<T$ (if Assumption B.3 ii) b) holds) or $\tau^{\eta}<T_{\eta}$ (if
Assumption B.3 ii) a) holds). ∎

###  B.2 ODE systems with negative feedback

In this section, we establish a convergence result for ODEs with random
coefficients and “nearly singular” driver. Specifically, we consider pairs of
continuously differentiable stochastic processes $(X^{\eta},Z^{\eta})$, which
satisfy the ODE system

$\begin{split}\dot{X}^{\eta}_{t}=&-A^{\eta}_{t}\big{(}X^{\eta}_{t}+B^{\eta}_{t}Z^{\eta}_{t}\big{)},\quad
X^{\eta}_{0}=x_{0},\\\ \dot{Z}^{\eta}_{t}=&\leavevmode\nobreak\
C_{t}X^{\eta}_{t}+D_{t}Z^{\eta}_{t},\quad Z^{\eta}_{0}=z_{0}\end{split}$
(B.11)

on some time interval $[0;S]$ for all $\eta>0$, where $A^{\eta}$, $B^{\eta}$,
$C$ and $D$ are continuous, adapted, real-valued stochastic processes. We
assume that the process $A^{\eta}$ converges in probability to infinity as
$\eta\to 0$, that the process $B^{\eta}$ converges to a process $B^{0}$ in
probability and that the processes $(X^{\eta},Z^{\eta})$ are uniformly bounded
in probability.

######  Assumption B.5.

There exists a continuous adapted process $B^{0}$, such that for all
$\varepsilon,\delta\in(0;\infty)$, there exists some
$\eta_{0}=\eta_{0}(\varepsilon,\delta)>0$ such that, for all
$\eta\in(0;\eta_{0}]$,

$\displaystyle\mathbb{P}\bigg{[}\inf_{t\in[0;S]}A^{\eta}_{t}\geq\frac{1}{\varepsilon},\sup_{t\in[0;S]}\big{|}B^{\eta}_{t}-B^{0}_{t}\big{|}\leq\varepsilon\bigg{]}\geq
1-\delta.$ (B.12)

Moreover, for all $\delta\in(0;1)$, there exists an $L>0$ such that, for all
$\eta>0$,

$\mathbb{P}\bigg{[}\sup_{t\in[0;S]}\Big{(}\big{|}X^{\eta}_{t}\big{|}+\big{|}Z^{\eta}_{t}\big{|}\Big{)}\leq
L\bigg{]}\geq 1-\delta.$ (B.13)

In terms of the process $B^{0}$, we define

$\begin{split}X^{0}_{t}&:=-B^{0}_{t}Z^{0}_{t},\\\
Z^{0}_{t}&:=z_{0}\exp\bigg{(}\int^{t}_{0}\big{(}-C_{s}B^{0}_{s}+D_{s}\big{)}\mathrm{d}s\bigg{)}.\end{split}$
(B.14)

Our goal is to prove that $(X^{\eta},Z^{\eta})\to(X^{0},Z^{0})$ as $\eta\to 0$
in a suitable sense. Since the initial condition of the processes $X^{\eta}$
does not vary with $\eta$ and since $A^{\eta}_{0}\uparrow\infty$, we cannot
expect convergence in $t=0$. Instead, we first prove that
$\frac{\dot{X}^{\eta}}{A^{\eta}}$ converges to $0$ on any compact subinterval
of $(0;S]$ with lower, respectively upper convergence near the origin.

######  Lemma B.6.

If Assumption B.5 holds and if $x_{0}+B^{0}_{0}z_{0}>0$, then for all
$\varepsilon,\delta\in(0;\infty)$, there exists
$\eta_{1}=\eta_{1}(\varepsilon,\delta)>0$ such that, for all
$\eta\in(0;\eta_{1}]$,

$\displaystyle\mathbb{P}\bigg{[}\sup_{t\in[0;S]}\frac{\dot{X}^{\eta}_{t}}{A^{\eta}_{t}}\leq\varepsilon,\inf_{t\in[\varepsilon;S]}\frac{\dot{X}^{\eta}_{t}}{A^{\eta}_{t}}\geq-\varepsilon\bigg{]}\geq
1-\delta.$

If $x_{0}+B^{0}_{0}z_{0}<0$, then the intervals $[0;S]$ and $[\varepsilon;S]$
in the above statement are to be swapped.

The mapping $\eta_{1}$ depends only on $B^{0}$, $C$, $D$, $x_{0}$, $z_{0}$,
and the mappings $(\varepsilon,\delta)\mapsto\eta_{0}(\varepsilon,\delta)$ and
$\delta\mapsto L$ introduced in Assumption B.5.

###### Proof.

Let $y_{0}:=x_{0}+B^{0}_{0}z_{0}$. We assume w.l.o.g. that $y_{0}>0$. Then
there exists $\alpha_{0}>0$ s.t.

$\displaystyle y_{0}-\alpha|z_{0}|>0\quad\mbox{for all }\alpha\leq\alpha_{0}.$

Let us now put

$\displaystyle M^{L,\eta}:=$
$\displaystyle\bigg{\\{}\sup_{t\in[0;S]}\Big{(}\big{|}B^{0}_{t}\big{|}+|C_{t}|+|D_{t}|+\big{|}X^{\eta}_{t}\big{|}+\big{|}Z^{\eta}_{t}\big{|}\Big{)}\leq
L\bigg{\\}},$ $\displaystyle M^{\alpha,\eta}:=$
$\displaystyle\bigg{\\{}\inf_{t\in[0;S]}A^{\eta}\geq\frac{1}{\alpha},\sup_{t\in[0;S]}\big{|}B^{\eta}_{t}-B^{0}_{t}\big{|}\leq\alpha\bigg{\\}},$
$\displaystyle M^{\beta,\nu}:=$
$\displaystyle\bigg{\\{}\sup_{s,t\in[0;S],|s-t|\leq\nu}\big{|}B^{0}_{s}-B^{0}_{t}\big{|}\leq\beta\bigg{\\}}.$