#  EXISTENCE, UNIQUENESS AND STRICT COMPARISON THEOREMS FOR BACKWARD
STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY RCLL MARTINGALES

Tianyang Nie$\,{}^{a}$111The research of T. Nie and M. Rutkowski was supported
by the DVC Research Bridging Support Grant Pricing of American and game
options in market with frictions. The work of T. Nie was supported by the
National Natural Science Foundation of China (No.
12022108,11971267,11831010,61961160732). and Marek Rutkowski$\,{}^{b,c}$

${}^{a\,}$School of Mathematics, Shandong University,
Jinan, Shandong 250100, China

${}^{b\,}$School of Mathematics and Statistics, University of Sydney
Sydney, NSW 2006, Australia

${}^{c\,}$Faculty of Mathematics and Information Science, Warsaw University of
Technology,
00-661 Warszawa, Poland

###### Abstract

Results on the existence, uniqueness and strict comparison for solutions to a
BSDE driven by a multi-dimensional RCLL martingale are established. The goal
is to develop a general multi-asset framework encompassing a wide spectrum of
nonlinear financial models with jumps, including as particular cases the
setups studied by Peng and Xu [27, 28] and Dumitrescu et al. [7] who dealt
with BSDEs driven by a one-dimensional Brownian motion and a purely
discontinuous martingale with a single jump.

Keywords: backward stochastic differential equation, RCLL martingale,
comparison theorem AMS Subject Classification: 60H10, 60H30, 91G30, 91G40

**footnotetext: E-mail<EMAIL_ADDRESS>(Tianyang NIE);
<EMAIL_ADDRESS>(Marek RUTKOWSKI).

##  1 Introduction

The goal of this paper is to reexamine and generalize results from papers by
Carbone et al. [4], Dumitrescu et al. [7], El Karoui and Huang [10], Nie and
Rutkowski [24] and Peng and Xu [27, 28] on the backward stochastic
differential equations (BSDEs) driven by multi-dimensional either continuous
([24], for the extended studies on BSDEs driven by continuous martingales, the
readers are referred to e.g. [22, 23] and the reference in [24])) or RCLL ([4,
7, 10, 27, 28]) martingales by proving several original results and providing
a comprehensive study of a particular class of BSDEs driven by a multi-
dimensional RCLL martingale. In particular, we establish the existence and
uniqueness theorem for solutions to a BSDE as well as a fairly general version
of the strict comparison theorem for solutions to BSDEs, which covers both the
classical comparison theorems driven by a Brownian motion (see, e.g., El
Karoui et al. [12]) and their extensions to BSDEs with an additional jump
martingale. The latter case was previously studied by Peng and Xu [27, 28] and
Dumitrescu et al. [7] who considered BSDEs driven by a combination of the
Wiener process and a single-jump compensated martingale associated with a
particular event, which can be interpreted as a source of an extraneous risk
(for instance, the risk of a catastrophe, an extremal weather event or a
bankruptcy of a company) and is characterized by the process of a stochastic
intensity of its occurrence. It is clear that the framework studied in the
present work includes the case of a single-jump martingale but it covers also
the case of several extraneous events occurring within a predetermined time
frame and, more generally, the case of an arbitrary RCLL martingale with the
predictable covariation satisfying Assumption 2.1.

The main practical motivation for this research is to provide an underpinning
for stochastic models of nonlinear financial markets with credit risk where
multi-dimensional discontinuous martingales appear in a natural way. In
particular, our results encompass market models with a single default event,
which were previously studied in Peng and Xu [27, 28] and Dumitrescu et al.
[7]. It should be stressed that in [27, 28] and [7], the authors examine a
special class of BSDEs driven by a one-dimensional Brownian motion and a
purely discontinuous martingale with a single jump of a unit size. In
contrast, we study BSDEs driven by an arbitrary multi-dimensional martingale
belonging to the class of multi-dimensional RCLL martingales satisfying
Assumption 2.1. We also postulate the predictable representation property of
the driving martingale, which is known to hold in some circumstances (see,
e.g., Kusuoka [21] or Jeanblanc and Le Cam [16]). Note, however, that this
assumption can be readily relaxed in the existence and uniqueness of solutions
theorem, provided that an additional orthogonal martingale term is introduced
in the definition of a solution to a BSDE (see, e.g., Carbone et al. [4] and
El Karoui and Huang [10]) and thus this possible extension is not presented
here.

Due to their generality, our results are capable of covering a much broader
spectrum of financial models, for instance, models with the counterparty
credit risk where several default times, and thus also multi-dimensional RCLL
martingales, appear in a natural way. By the same token, results from this
work can be used to study market models where asset prices are governed by
noise processes with jumps. Applications of our results on BSDEs and reflected
BSDEs driven by RCLL martingales to the valuation, hedging and exercising of
American and game options are presented in the follow-up works by Kim et al.
[19, 20]. Finally, it is also worth mentioning that several classes of BSDEs
driven by a Brownian motion and a martingale random measure were studied by,
among others, Barles et al. [1], El Karoui et al. [11], Jeanblanc et al. [17,
18], Quenez and Sulem [30], Royer [31] and Tang and Li [32]. We stress that
their results do not cover the class of BSDEs driven by RCLL martingales
considered in the present work.

After our research was completed, we learnt about an excellent recent paper by
Papapantaleon et al. [26] where the authors examined well-posedness of BSDEs
driven by a general martingale in a possibly stochastically discontinuous
filtration. In particular, results from [26] are capable of covering BSDEs in
both continuous- and discrete-time frameworks, unlike all other papers on
continuous-time BSDEs driven by discontinuous martingales (and possibly also a
Poisson random measure). We acknowledge that our goals were different and by
far less ambitious since we focused on a particular setup, which is well-
adapted to direct applications in the study of nonlinear financial markets
with credit risk where the intensity-based approach to modeling of default
times is ubiquitous (see, e.g., Bielecki et al. [3]). Although our setup is
similar to that of Carbone et al. [4], we contend that the original
contributions of the present work are nontrivial and of tangible practical
importance. For the reader’s convenience, we compare our methods with those
from [4, 26] and we argue that our results cannot be obtained from results
established in either [4] or [26].

Firstly, we apply directly Itô’s formula and construct the contraction mapping
by using the supremum norm for the component $Y$ of a solution to a BSDE (or
the process $Y-D$ in a slightly more general setup of BSDE (2.2)). In Section
3.6 of [26] and Section 3 of [4], the authors also apply Itô’s formula to the
appropriately weighted $L^{2}$-type norm (see page 44 in [26] and the proof of
Propostion 3.1 in [4]) but they construct the contraction mapping by using
this weighted $L^{2}$-type norm for $Y$ and thus they cannot guarantee that
$Y$ is an RCLL process and, furthermore, are unable to ensure that the
uniqueness of a solution to a BSDE holds in the sense of indistinguishability
of stochastic processes (see, e.g., Theorem 3.23 in [26] and Theorem 3.1 in
[4]). These manifestly technical, but nevertheless important in practical
applications, shortcomings stem from the fact that the modification procedure
for the classical BSDE (see, e.g., [12]) is no longer applicable when the
process $\langle M\rangle$ (the predictable covariation process of the
martingale $M$) is not absolutely continuous with respect to the Lebesgue
measure.

Secondly, and more importantly, we give in Proposition 5.1 an explicit
solution to the linear BSDE and subsequently use its form to prove the strict
comparison theorem, which is a crucial result for financial applications, most
notably, the arbitrage-free pricing of European, American and game options in
nonlinear markets (see [2, 19, 20]). Carbone et al. [4] established the
comparison theorem under some assumptions which are not easy to verify (see
Theorem 2.2 and Lemma 2.2 in [4]) and they studied a BSDE with the generator
term involving integration w.r.t. $\langle M\rangle$, as opposed to the
integration w.r.t. a process $Q$ as in the BSDE (2.3). Despite the fact that
the integration w.r.t. $Q$ entails additional difficulties, we were able to
establish the strict comparison theorem by first obtaining an explicit
expression for a unique solution to the linear BSDE with integration w.r.t.
$Q$, which is also different from [4]. It should also be noticed that using
the method of [26] it is possible to establish the comparison theorem under
rather restrictive assumptions (see Theorem 3.25 in [26]) but not the strict
comparison theorem obtained in the present work (see Theorem 6.1).

The paper is organized as follows. We start by recalling some definitions from
[4, 10, 24] pertaining to the backward stochastic differential equations
driven by a multi-dimensional martingale. For the sake of conciseness, we
refer the interested reader to [24] for further references and comments on
various Lipschitz-type conditions for generators of BSDEs appearing in
financial applications. First, in Proposition 3.1 we obtain the a priori
estimates for a solution to a BSDE. Next, in Theorem 4.1, we prove the
existence and uniqueness of a solution to a BSDE under the assumption that its
generator is uniformly $m$-Lipschitz continuous. Proposition 5.1 furnishes an
explicit representation for a unique solution to the linear BSDE. Finally, in
Theorem 6.1 we establish the strict comparison property of solutions to the
BSDE (2.2). Let us conclude the introduction by mentioning that as the
contiuation of this work, reflected BSDEs and doubly reflected BSDEs are
investigated in [25].

##  2 BSDEs Driven by RCLL Martingales

We assume that we are given a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions
of right-continuity and completeness. Moreover, the initial $\sigma$-field
$\mathcal{F}_{0}$ is assumed to be trivial. For any two processes, say $X$ and
$Y$, the equality $X=Y$ means that
$\mathbb{P}(X_{t}=Y_{t},\,\forall\,t\in[0,T])=1$. By convention, we set
$X_{0-}=0$ for any stochastic process $X$ and thus $\Delta
X_{0}:=X_{0}-X_{0-}=X_{0}$ is the jump of $X$ at time 0. Finally, the integral
$\int_{s}^{t}X_{u}\,dY_{u}$ is interpreted as $\int_{]s,t]}X_{u}\,dY_{u}$
where $]s,t]=\\{u\in[0,T]:s<u\leq t\\}$.

Let $M=((M^{1}_{t},M^{2}_{t},\ldots,M^{n}_{t})^{\ast},\,t\in[0,T])$ be an
$n$-dimensional, square-integrable martingale defined on a filtered
probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$. Since the
filtration $\mathbb{F}$ is assumed to be right-continuous, an RCLL
modification of any $\mathbb{F}$-martingale is known to exist and thus it is
assumed that $M$ is an RCLL process. We denote by $\langle M\rangle$ (resp.
$[M]$) the predictable covariation process (resp. covariation process) of $M$
meaning that the process $\langle M\rangle$ (resp. $[M]$) takes values in
$\mathbb{R}^{n\times n}$ and the $(j,k)$th entry of the random matrix $\langle
M\rangle_{t}$ (resp. $[M]$) equals $\langle M^{j},M^{k}\rangle_{t}$ (resp.
$[M^{j},M^{k}]$). As in [4, 10, 24], we henceforth work under the following
standing assumption regarding the predictable covariation of $M$.

######  Assumption 2.1.

There exists an $\mathbb{R}^{n\times n}$-valued, $\mathbb{F}$-predictable
process $m$ and an $\mathbb{F}$-adapted, continuous, nondecreasing process $Q$
with $Q_{0}=0$ such that, for all $t\in[0,T]$,

$\langle M\rangle_{t}=\int_{0}^{t}m_{u}m_{u}^{\ast}\,dQ_{u}.$ (2.1)

In addition, the process $Q$ is bounded: there exists a constant $C_{Q}$ such
that $Q_{t}\leq C_{Q}$ for all $t\in[0,T]$.

The postulated continuity of $Q$ entails that the process $\langle M\rangle$
is continuous as well. This property is known to be satisfied when $M$ is an
$\mathbb{F}$-quasi-left-continuous process, meaning that it does not have
discontinuities at $\mathbb{F}$-predictable stopping times (see Chapter VI in
He et al. [14] or Chapter 4 in Protter [29]). In particular, it holds when the
filtration $\mathbb{F}$ is assumed to be quasi-left-continuous (as was
postulated in Carbone et al. [4]), that is, if
$\mathcal{F}_{\tau}=\mathcal{F}_{\tau-}$ for every $\mathbb{F}$-predictable
stopping time $\tau$. Indeed, the latter property is known to be valid if and
only if any uniformly integrable $\mathbb{F}$-martingale is a quasi-left-
continuous process. In fact, if the filtration $\mathbb{F}$ is quasi-left-
continuous, then $\mathbb{F}$-martingales jump only at $\mathbb{F}$-totally
inaccessible stopping times.

The boundedness $Q$ is easy to be satisfied, for more details, see e.g. Remark
2.1 in Nie and Rutkowski [24] or Section 2.1 in Morlais [23]. Let us also
mention that it looks that the boundedness of $Q$ is a useful and, in some
sense, a necessary condition. Indeed, as we explained in the introduction, to
show that the component $Y$ of a solution to a BSDE is an RCLL process and
ensure that the BSDE holds in the sense of indistinguishability of stochastic
processes, the contraction mapping based on the weighted $L^{2}$-type norm for
$Y$ (see page 44 in [26] and proof of Proposition 3.1 in [4]) is not
sufficient. Hence we construct the contraction mapping by using the supremum
norm for $Y$ and for that purpose the boundedness of $Q$ is needed (see the
proof of Theorem 4.1).

For a positive semi-definite matrix $a$, we denote by $a^{\frac{1}{2}}$ the
unique square root of $a$, that is, $a^{\frac{1}{2}}a^{\frac{1}{2}}=a$. Recall
that there exists an orthogonal matrix $O$ and a diagonal matrix $b$ with
nonnegative diagonal entries such that $a=O^{\ast}bO$. The square root of $b$
is also a diagonal matrix, denoted by $b^{\frac{1}{2}}$, with diagonal entries
equal to square roots of diagonal entries of $b$. Then we set
$a^{\frac{1}{2}}=O^{\ast}b^{\frac{1}{2}}O$. Moreover, if $a$ is positive
definite, then the inverse of $a^{\frac{1}{2}}$, denoted as
$a^{-\frac{1}{2}}$, is well defined. It is known that $m_{u}m_{u}^{\ast}$ is a
square, positive semi-definite matrix, so that
$(m_{u}m_{u}^{\ast})^{\frac{1}{2}}$ is well defined. If $m_{u}m_{u}^{\ast}$ is
positive definite, then $m_{u}m_{u}^{\ast}$ is invertible, and thus we can
also define $n_{u}:=(m_{u}m_{u}^{\ast})^{-\frac{1}{2}}$.

Regarding the symmetry of the matrix-valued process $m$ appearing in (2.1),
observe that condition (2.1) with an $\mathbb{R}^{n\times n}$-valued,
$\mathbb{F}$-predictable process $m$ is equivalent to $\langle
M\rangle_{t}=\int_{0}^{t}\bar{m}_{u}\bar{m}_{u}^{\ast}\,dQ_{u}$ with a
symmetric $\mathbb{R}^{n\times n}$-valued, $\mathbb{F}$-predictable process
$\bar{m}$ since it suffices to take
$\bar{m}_{u}=(m_{u}m_{u}^{\ast})^{\frac{1}{2}}$. Hence we may and do assume,
without loss of generality, that the process $m$ takes values in the space of
symmetric matrices so that
$m_{u}=m^{\ast}_{u}=(m_{u}m_{u}^{\ast})^{\frac{1}{2}}$.

Let $\|\cdot\|$ stand for the Euclidean norm in $\mathbb{R}^{n}$. We denote by
${\mathcal{L}}^{2}(M)$ the space of all $\mathbb{R}^{n}$-valued,
$\mathbb{F}$-predictable processes $Z$ with the pseudo-norm
$\|\cdot\|_{{\mathcal{L}}^{2}(M)}$ given by

$\displaystyle\|Z\|_{{\mathcal{L}}^{2}(M)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}\|m_{t}Z_{t}\|^{2}\,dQ_{t}\bigg{]}<\infty.$

We also make the following standing assumption, which provides a natural
underpinning for the existence of a solution the BSDE (2.2).

######  Assumption 2.2.

The martingale $M$ has the predictable representation property under
$\mathbb{P}$ with respect to the filtration $\mathbb{F}$, in the sense that
for any real-valued, square-integrable $\mathbb{F}$-martingale $N$ with
$N_{0}=0$ there exists a process $Z\in{\mathcal{L}}^{2}(M)$ such that
$N_{t}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}$. Moreover, the uniqueness of $Z$ in
$\|\cdot\|_{{\mathcal{L}}^{2}(M)}$ holds, that is, if
$N_{t}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}=\int_{0}^{t}\widetilde{Z}^{*}_{u}\,dM_{u}$,
then $\|Z-\widetilde{Z}\|_{{\mathcal{L}}^{2}(M)}=0$.

######  Remark 2.1.

Assumption 2.2 is known to be satisfied when $M$ is either a Wiener process or
a compensated Poisson process. Kusuoka [21] has shown that Assumption 2.2 is
valid for some special choice of $\mathbb{F}$ and $M$, which was made in [7,
27, 28] (see also Section 3.3 in Bielecki et al. [3], Theorem 2.1 in Jeanblanc
and Le Cam [16] or Assumption A.1 and Example 1 in Jeanblanc et al. [17, 18]).
If Assumption 2.2 is relaxed, then the results of this paper can be
generalized by combining the methods used here with the arguments from [4, 10]
provided that we assume, in addition, that the filtration $\mathbb{F}$ is
quasi-left-continuous. Then the extended martingale representation property of
$M$ holds, in the sense that any real-valued, square-integrable
$\mathbb{F}$-martingale $N$ can be represented as
$N_{t}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}+L_{t}$ where $L$ is a square-integrable
$\mathbb{F}$-martingale orthogonal to $M$.

For a fixed horizon date $T>0$, we consider the following backward stochastic
differential equation (BSDE) on $[0,T]$ with data $(g,\eta,D)$

$\left\\{\begin{array}[c]{ll}dY_{t}=-g(t,Y_{t},Z_{t})\,dQ_{t}+Z_{t}^{\ast}\,dM_{t}+dD_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ Y_{T}=\eta,\end{array}\right.$ (2.2)

where $D$ is a given real-valued, $\mathbb{F}$-adapted process or, more
explicitly, for every $t\in[0,T]$,

$\displaystyle
Y_{t}=\eta+\int_{t}^{T}g(t,Y_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}-(D_{T}-D_{t})$
(2.3)

where, as usual, equality (2.3) is assumed to be satisfied $\mathbb{P}$-a.s..
The next definition states the measurability properties of the generator $g$.

######  Definition 2.1.

A generator
$g:\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}$
is a
$\mathcal{F}\otimes\mathcal{B}([0,T])\otimes\mathcal{B}(\mathbb{R})\otimes\mathcal{B}(\mathbb{R}^{n})$-measurable
function such that the process $g(\cdot,\cdot,y,z)$ is
$\mathbb{F}$-progressively measurable, for every
$(y,z)\in\mathbb{R}\times\mathbb{R}^{n}$.

We denote by $\mu_{Q}$ the Doléans measure of the process $Q$ on the space
$\Omega\times[0,T]$ endowed with the product $\sigma$-algebra
$\mathcal{F}_{T}\otimes\mathcal{B}([0,T])$ so that
$\mu_{Q}(A)={\mathbb{E}}\big{[}\int_{0}^{T}\mathds{1}_{A}(\omega,t)\,dQ_{t}(\omega)\big{]}$
for every $A\in\mathcal{F}_{T}\otimes\mathcal{B}([0,T])$. We first recall the
Lipschitz condition, which is frequently employed in the existing literature
on BSDEs driven by a Brownian motion.

######  Definition 2.2.

A generator $g$ is uniformly Lipschitz continuous if there exists a constant
$L$ such that, for $\mu_{Q}$-almost every $(\omega,t)$, for all
$y_{1},y_{2}\in\mathbb{R},\,z_{1},z_{2}\in\mathbb{R}^{n}$,

$|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq
L\big{(}|y_{1}-y_{2}|+\|z_{1}-z_{2}\|\big{)}.$ (2.4)

Let ${\mathcal{S}}^{2}$ stand for the space of all real-valued, RCLL,
$\mathbb{F}$-adapted processes endowed with the norm
$\|\cdot\|_{{\mathcal{S}}^{2}}$ given by

$\displaystyle\|X\|_{{\mathcal{S}}^{2}}^{2}:={\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}X^{2}_{t}\bigg{]}<\infty.$

We denote by ${\mathcal{H}}^{2}(Q)$ the space of equivalence classes of all
real-valued, $\mathbb{F}$-progressively measurable processes $X$ with the
pseudo-norm $\|\cdot\|_{{\mathcal{H}}^{2}(Q)}$ given by

$\displaystyle\|X\|_{{\mathcal{H}}^{2}(Q)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}X^{2}_{t}\,dQ_{t}\bigg{]}<\infty.$

We observe that ${\mathcal{S}}^{2}\subset{\mathcal{H}}^{2}(Q)$ and, for
brevity, we denote by $\mathbb{S}^{2}$ the product space
${\mathcal{S}}^{2}\times{\mathcal{L}}^{2}(M)$ with the following pseudo-norm,
for every $(X,Z)\in{\mathcal{S}}^{2}\times{\mathcal{L}}^{2}(M)$,

$\displaystyle\|(X,Z)\|_{\mathbb{S}^{2}}=\|X\|_{{\mathcal{S}}^{2}}+\|Z\|_{{\mathcal{L}}^{2}(M)}.$

By the same token, we introduce the space
$\mathbb{H}^{2}:={\mathcal{H}}^{2}(Q)\times{\mathcal{L}}^{2}(M)$ endowed with
the natural pseudo-norm

$\displaystyle\|(X,Z)\|_{\mathbb{H}^{2}}=\|X\|_{{\mathcal{H}}^{2}(Q)}+\|Z\|_{{\mathcal{L}}^{2}(M)}.$

As usual, $L^{2}({\mathcal{F}}_{T})$ stands for the space of all real-valued,
$\mathcal{F}_{T}$-measurable random variables $\eta$ such that
$\|\eta\|_{L^{2}({\mathcal{F}}_{T})}^{2}={\mathbb{E}}(\eta^{2})<\infty$.

In the study of the classical BSDEs driven by a Brownian motion (also with
jumps generated by a Poisson random measure), it suffices to take $Q_{t}=t$.
Moreover, in order to ensure the existence and uniqueness of a solution
$(Y,Z)$ it is common to postulate that a generator $g$ is uniformly Lipschitz
continuous with respect to the variables $y$ and $z$. However, it was observed
that the uniform Lipschitz condition (2.4) is no longer suitable when dealing
with BSDEs with a single jump, which was aimed to represent the occurrence of
the default event, and thus condition (2.4) has been modified to explicitly
account for the intensity process $\lambda$. For instance, Theorem 3.1 in Peng
and Xu [27, 28] and Proposition 2 in Dumitrescu et al. [7] (see also
Proposition 2.3 in Dumitrescu et al. [8]) hinge on the postulate that a
generator satisfies the following condition

$|g(t,y_{1},z_{1},k_{1})-g(t,y_{2},z_{2},k_{2})|\leq
L\big{(}|y_{1}-y_{2}|+\|z_{1}-z_{2}\|+\sqrt{\lambda_{t}}|k_{1}-k_{2}|\big{)}$
(2.5)

where the variables $z_{1},z_{2}$ (respectively, $k_{1},k_{2}$) correspond to
the one-dimensional Brownian motion (respectively, the compensated martingale
of the jump process). A generator $g(t,y,z,k)$ satisfying condition (2.5) and
such that the process $g_{t}:=g(t,0,0,0)$ belongs to ${\mathcal{H}}^{2}(Q)$ is
called the $\lambda$-admissible driver in [7]

More generally, in existing papers concerned with BSDEs driven by a multi-
dimensional martingale (see, e.g., [4, 10, 24]) a generator is typically
postulated to satisfy some kind of the $m$-Lipschitz condition where $m$ is
the process introduced in Assumption 2.1. In particular, the BSDE driven by a
multi-dimensional continuous martingale and with a uniformly $m$-Lipschitz
continuous generator was studied in [24] where the authors introduced also
certain alternative conditions, which are more adequate when the process $m$
is unbounded. In the present work, we focus on BSDEs with generator satisfying
the following regularity condition.

######  Definition 2.3.

A generator $g$ is uniformly $m$-Lipschitz continuous if there exists a
constant $\widehat{L}>0$ such that, for $\mu_{Q}$-almost every $(\omega,t)$,
and all $y_{1},y_{2}\in\mathbb{R},\,z_{1},z_{2}\in\mathbb{R}^{n}$,

$|g(t,y_{1},z_{1})-g(t,y_{2},z_{2})|\leq\widehat{L}\big{(}|y_{1}-y_{2}|+\|m_{t}(z_{1}-z_{2})\|\big{)}.$
(2.6)

######  Remark 2.2.

We note that condition (2.5), which is employed in Peng and Xu [27, 28] and
Dumitrescu et al. [7], can be seen as a special case of (2.6). Indeed, suppose
that $M^{1}:=W$ is a Brownian motion and $M^{2}_{t}:=\mathds{1}_{\\{\tau\leq
t\\}}-\int_{0}^{t}\lambda_{u}\,du$ is the compensated martingale of the
indicator process of a strictly positive $\mathbb{F}$-stopping time $\tau$
with the $\mathbb{F}$-intensity process $\lambda$ (note that, as in [7], we
use here the convention that the intensity process $\lambda$ is
$\mathbb{F}$-progressively measurable and thus it necessarily vanishes after
$\tau$). Then we have $\langle M^{1}\rangle_{t}=\langle W\rangle_{t}=t$ and
$\langle M^{2}\rangle_{t}=\int_{0}^{t}\lambda_{u}\,du$ since $M^{2}$ is a
purely discontinuous martingale so that, from Theorem 4.52 in Jacod and
Shiryaev [15], we obtain $[M^{2}]_{t}=\sum_{u\leq t}(\Delta
M^{2}_{u})^{2}=\mathds{1}_{\\{\tau\leq t\\}}$. Moreover, $M^{1}$ is a
continuous martingale and $M^{2}$ is a purely discontinuous martingale and
thus, by Corollary 4.55 in [15], we have $[M^{1},M^{2}]_{t}=\langle
M^{1},M^{2}\rangle_{t}=0$. Hence, if we set $Q_{t}=t$, then (2.5) can be
identified with a particular instance of condition (2.6), which shows that a
$\lambda$-admissible driver satisfies Definition 2.3.

We adopt the following definition of a solution to the BSDE (2.2).

######  Definition 2.4.

A solution to the BSDE (2.2) with data $(g,\eta,D)$ is a pair of stochastic
processes $(Y,Z)\in\mathbb{H}^{2}$ such that equality (2.2) is satisfied
pathwise, in the sense that

$\displaystyle\mathbb{P}\bigg{(}Y_{t}=\eta+\int_{t}^{T}g(u,Y_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}-(D_{T}-D_{t}),\,\forall\,t\in[0,T]\bigg{)}=1.$

We say that the uniqueness of a solution to (2.2) holds if for any two
solutions $(Y,Z)$ and $(Y^{\prime},Z^{\prime})$ to (2.2) we have
$\|(Y,Z)-(Y^{\prime},Z^{\prime})\|_{\mathbb{H}^{2}}=0$.

We henceforth study BSDEs with generators satisfying Definition 2.3 and data
$(g,\eta,D)$ satisfying the following assumption.

######  Assumption 2.3.

The triplet $(g,\eta,D)$ is such that: (i) the generator $g$ is uniformly
$m$-Lipschitz continuous and the process $g(\cdot,0,0)$ belongs to
${\mathcal{H}}^{2}(Q)$, (ii) the random variable $\eta$ belongs to
$L^{2}({\mathcal{F}}_{T})$, (iii) the process $D$ belongs to
${\mathcal{H}}^{2}(Q)$ with $D_{T}\in L^{2}({\mathcal{F}}_{T})$.

######  Remark 2.3.

Under Assumption 2.3, Definition 2.4 can be restated as follows: a pair
$(Y,Z)\in\mathbb{H}^{2}$ is a solution to the BSDE (2.2) if the process $Y-D$
is RCLL and, for every $t\in[0,T]$, equality (2.3) holds $\mathbb{P}$-a.s.. In
fact, we prove in Theorem 4.1 that, under Assumptions 2.1–2.3, there exists a
solution $(Y,Z)$ to the BSDE (2.2) such that $(Y-D,Z)$ belongs to the space
$\mathbb{S}^{2}$. To this end, we first establish some useful a priori
estimates in Proposition 3.1.

##  3 A Priori Estimates

We first derive some useful a priori estimates for solutions to BSDEs.

######  Proposition 3.1.

Let Assumptions 2.1–2.3 be satisfied by the martingale $M$ and the triplets
$(g^{l},\eta^{l},D^{l})$ for $l=1,2$. For $l=1,2$, let $(Y^{l},Z^{l})$ be a
solution to the BSDE

$\left\\{\begin{array}[c]{ll}dY_{t}^{l}=-g^{l}(t,Y^{l}_{t},Z^{l}_{t})\,dQ_{t}+Z^{l,\ast}_{t}\,dM_{t}+dD^{l}_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ Y_{T}^{l}=\eta^{l},\end{array}\right.$ (3.1)

and let
$y:=Y^{1}-Y^{2},\,z:=Z^{1}-Z^{2},\,D:=D^{1}-D^{2},\,\eta:=\eta^{1}-\eta^{2}$
and $g:=g^{1}-g^{2}$. Then there exists a constant $C\geq 0$ such that

$\|y-D\|_{{\mathcal{S}}^{2}}^{2}+\|y\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|z\|_{{\mathcal{L}}^{2}(M)}^{2}\leq
CJ$ (3.2)

where

$\displaystyle J:=\|\eta-
D_{T}\|^{2}_{L^{2}({\mathcal{F}}_{T})}+\|D\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|g(t,Y^{2},Z^{2})\|_{{\mathcal{H}}^{2}(Q)}^{2}<\infty.$

Proof. We write $\widetilde{Y}^{i}=Y^{i}-D^{i}$, $\widetilde{Z}^{i}=Z^{i}$,
$\widetilde{\eta}^{i}=\eta^{i}-D_{T}^{i}$,
$\widetilde{g}^{i}(t,y,z):=g^{i}(t,y+D_{t}^{i},z)$,
$\widetilde{y}:=\widetilde{Y}^{1}-\widetilde{Y}^{2},\,\widetilde{z}:=\widetilde{Z}^{1}-\widetilde{Z}^{2},\,D:=D^{1}-D^{2}$
and $\widetilde{\eta}:=\widetilde{\eta}^{1}-\widetilde{\eta}^{2}$. Then we
have

$\left\\{\begin{array}[c]{ll}d\widetilde{y}_{t}=-\delta_{t}\,dQ_{t}+\widetilde{z}^{*}_{t}\,dM_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
\widetilde{y}_{T}=\widetilde{\eta},\end{array}\right.$ (3.3)

where
$\delta_{t}:=\widetilde{g}^{1}(t,\widetilde{Y}^{1}_{t},\widetilde{Z}^{1}_{t})-\widetilde{g}^{2}(t,\widetilde{Y}^{2}_{t},\widetilde{Z}^{2}_{t})$.
Since $[e^{\beta Q},\widetilde{y}^{2}]=0$ by virtue of Proposition 4.49 in
Chapter I of [15], the Itô integration by parts formula gives

$\displaystyle d(e^{\beta
Q_{t}}\widetilde{y}^{2}_{t})=\widetilde{y}^{2}_{t}\,de^{\beta Q_{t}}+e^{\beta
Q_{t}}\,d\widetilde{y}^{2}_{t}=\widetilde{y}^{2}_{t}\beta e^{\beta
Q_{t}}\,dQ_{t}+e^{\beta Q_{t}}\,d\widetilde{y}^{2}_{t}$

where

$\displaystyle
d\widetilde{y}^{2}_{t}=2\widetilde{y}_{t-}\,d\widetilde{y}_{t}+d[\widetilde{y}]_{t}=2\widetilde{y}_{t-}\,d\widetilde{y}_{t}+dN_{t}+d\langle\widetilde{y}\rangle_{t}$

and the process $N:=[\widetilde{y}]-\langle\widetilde{y}\rangle$ is an
$\mathbb{F}$-martingale. In view of (3.3), we have

$\displaystyle\widetilde{y}_{t-}\,d\widetilde{y}_{t}=-\widetilde{y}_{t}\delta_{t}\,dQ_{t}+\widetilde{y}_{t-}z^{*}_{t}\,dM_{t}$

and $d\langle\widetilde{y}\rangle_{t}=\|m_{t}\widetilde{z}_{t}\|^{2}\,dQ_{t}$.
By integrating from $t$ to $T$ and taking the conditional expectation with
respect to $\mathcal{F}_{t}$, we obtain

$\begin{split}&e^{\beta
Q_{t}}\widetilde{y}^{2}_{t}+\beta\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\widetilde{y}^{2}_{u}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}\widetilde{z}_{u}\|^{2}\,dQ_{u}\bigg{]}\\\
&={\mathbb{E}}_{t}\Big{[}e^{\beta
Q_{T}}\widetilde{\eta}^{2}\Big{]}+2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\widetilde{y}_{u}\delta_{u}\,dQ_{u}\bigg{]}.\end{split}$ (3.4)

Let us denote

$\displaystyle\widetilde{g}_{t}:=\widetilde{g}^{1}(t,\widetilde{Y}^{2}_{t},\widetilde{Z}^{2}_{t})-\widetilde{g}^{2}(t,\widetilde{Y}^{2}_{t},\widetilde{Z}^{2}_{t})=g^{1}(t,Y^{2}_{t}-D^{2}_{t}+D^{1}_{t},Z^{2}_{t})-g^{2}(t,Y^{2}_{t},Z^{2}_{t}).$

Since
$|\delta_{t}|\leq\widehat{L}_{1}(|\widetilde{y}_{t}|+\|m_{t}\widetilde{z}_{t}\|)+|\widetilde{g}_{t}|$,
we deduce from (3.4) that

$\displaystyle e^{\beta Q_{t}}\widetilde{y}^{2}_{t}$
$\displaystyle+\beta\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\widetilde{y}^{2}_{u}\,dQ_{u}\bigg{]}+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}\widetilde{z}_{u}\|^{2}\,dQ_{u}\bigg{]}$
$\displaystyle\leq{\mathbb{E}}_{t}\Big{[}e^{\beta
Q_{T}}\widetilde{\eta}^{2}\Big{]}+2\widehat{L}_{1}\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\widetilde{y}^{2}_{u}\,dQ_{u}\bigg{]}+2\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|\widetilde{y}_{u}|\big{(}\widehat{L}_{1}\|m_{u}\widetilde{z}_{u}\|+|\widetilde{g}_{u}|\big{)}\,dQ_{u}\bigg{]}$
$\displaystyle\leq{\mathbb{E}}_{t}\Big{[}e^{\beta
Q_{T}}\widetilde{\eta}^{2}\Big{]}+(2\widehat{L}_{1}+2\widehat{L}_{1}^{2}+1)\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\widetilde{y}^{2}_{u}\,dQ_{u}\bigg{]}$
$\displaystyle+{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\Big{(}\frac{1}{2}\|m_{u}\widetilde{z}_{u}\|^{2}+|\widetilde{g}_{u}|^{2}\big{)}\Big{)}\,dQ_{u}\bigg{]}.$

We now take $\beta=2\widehat{L}_{1}+2\widehat{L}_{1}^{2}+2$ and rearrange the
above equation to obtain

$\displaystyle e^{\beta
Q_{t}}\widetilde{y}^{2}_{t}+\frac{1}{2}\,{\mathbb{E}}_{t}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}\widetilde{z}_{u}\|^{2}\,dQ_{u}\bigg{]}\leq{\mathbb{E}}\big{(}e^{\beta
Q_{T}}\widetilde{\eta}^{2}\big{)}+{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{t}}|\widetilde{g}_{t}|^{2}\,dQ_{t}\bigg{]}.$

Then, recalling also that $Q$ is bounded and taking $t=0$, we obtain the
existence of a constant $C\geq 0$ (note that the value of $C$ may change from
place to place) such that

$\displaystyle\|\widetilde{y}\|^{2}_{{\mathcal{S}}^{2}}+\|\widetilde{y}\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|\widetilde{z}\|_{{\mathcal{L}}^{2}(M)}^{2}\leq
C\bigg{[}{\mathbb{E}}\big{(}\widetilde{\eta}^{2}\big{)}+{\mathbb{E}}\bigg{(}\int_{0}^{T}|\widetilde{g}_{t}|^{2}\,dQ_{t}\bigg{)}\bigg{]}$

where we have also used the property that $\widetilde{y}$ is an RCLL process.
Noticing that $|\widetilde{g}_{t}|^{2}\leq
2|g(t,Y_{t}^{2},Z_{t}^{2})|^{2}+2\widehat{L}_{1}^{2}|D_{t}|^{2}$, we get

$\displaystyle\|\widetilde{y}\|^{2}_{{\mathcal{S}}^{2}}+\|\widetilde{y}\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|\widetilde{z}\|_{{\mathcal{L}}^{2}(M)}^{2}\leq
CJ,$

which in turn implies that (recalling that $\widetilde{y}=y-D$ and
$\widetilde{z}=z$)

$\displaystyle\|y-D\|^{2}_{{\mathcal{S}}^{2}}+\|y\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|z\|_{{\mathcal{L}}^{2}(M)}^{2}\leq
CJ.$

Finally, it is easy to check that $J$ is finite. $\Box$

######  Remark 3.1.

For later use, it is worth noting that (3.2) implies that if
$|\eta^{1}-\eta^{2}|\leq\varepsilon$ for some positive constant $\varepsilon$,
then the solutions $Y^{1}$ and $Y^{2}$ to the BSDE (2.2) with terminal
conditions $\eta^{1}$ and $\eta^{2}$, respectively, satisfy
$|Y^{1}_{0}-Y^{2}_{0}|\leq C\varepsilon$ for some constant $C$.

##  4 Existence and Uniqueness Theorem for BSDEs

To establish the existence and uniqueness of solutions to the BSDE (2.2), it
is useful to introduce several auxiliary normed spaces of stochastic processes
indexed by a parameter $\beta\geq 0$. First, for any fixed $\beta\geq 0$, we
denote by ${\mathcal{L}}^{2}_{\beta}(M)$ the class of all
$\mathbb{R}^{n}$-valued, $\mathbb{F}$-predictable processes $Z$ with the
pseudo-norm $\|\cdot\|_{{\mathcal{L}}^{2}_{\beta}(M)}$ given by

$\displaystyle\|Z\|_{{\mathcal{L}}^{2}_{\beta}(M)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{t}}\|m_{t}Z_{t}\|^{2}\,dQ_{t}\bigg{]}<\infty.$

Hence ${\mathcal{L}}^{2}(M)=\mathcal{L}^{2}_{0}(M)$ and the pseudo-norms
$\|\cdot\|_{{\mathcal{L}}^{2}(M)}$ and
$\|\cdot\|_{{\mathcal{L}}^{2}_{\beta}(M)}$ are equivalent for every $\beta>0$
since the nondecreasing, nonnegative process $Q$ is bounded.

Second, we denote by ${\mathcal{H}}^{2}_{\beta}(Q)$ the space of all real-
valued, $\mathbb{F}$-progressively measurable processes $X$ with the pseudo-
norm $\|\cdot\|_{{\mathcal{H}}^{2}_{\beta}(Q)}$ given by

$\displaystyle\|X\|_{{\mathcal{H}}^{2}_{\beta}(Q)}^{2}:={\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{t}}X^{2}_{t}\,dQ_{t}\bigg{]}<\infty.$

Third, we introduce the space ${\mathcal{S}}^{2}_{\beta}$ of all real-valued,
RCLL, $\mathbb{F}$-adapted processes $X$ with the norm
$\|\cdot\|_{\mathcal{S}_{\beta}^{2}}$ given by

$\displaystyle\|X\|_{{\mathcal{S}}^{2}_{\beta}}^{2}:={\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}X^{2}_{t}\big{)}\bigg{]}<\infty.$

It is clear that ${\mathcal{S}}^{2}=\mathcal{S}_{0}^{2}$ and the norms
$\|\cdot\|_{{\mathcal{S}}^{2}}$ and $\|\cdot\|_{{\mathcal{S}}^{2}_{\beta}}$
are equivalent for every $\beta>0$.

Finally, for a fixed $\beta\geq 0$, we denote by $\mathbb{S}^{2}_{\beta}$ the
Banach space
$({\mathcal{S}}^{2}_{\beta}\times{\mathcal{L}}^{2}_{\beta}(M),\|\cdot\|_{\beta})$
where the pseudo-norm $\|\cdot\|_{\beta}$ is given by

$\displaystyle\|(w,v)\|^{2}_{\beta}:=\|w\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}$

and we note that $\mathbb{S}^{2}=\mathbb{S}^{2}_{0}$. We stress that, in view
of Assumption 2.1, the class $\mathbb{S}^{2}_{\beta}$ of stochastic processes
is independent of the choice of $\beta\geq 0$ and thus the set
$\mathbb{S}^{2}_{\beta}$ can be formally identified with $\mathbb{S}^{2}$,
although the pseudo-norm $\|\cdot\|_{\beta}$ depends on $\beta$ and thus it is
advantageous to consider a whole family of pseudo-norms when searching for a
solution to the BSDE (2.2) through a contraction mapping on
$\mathbb{S}^{2}_{\beta}$ for some $\beta>0$.

Notice that equation (2.3) can also be rewritten as follows

$\displaystyle Y_{t}-D_{t}=\eta-
D_{T}+\int_{t}^{T}\widetilde{g}(t,Y_{u}-D_{u},Z_{u})\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}$

where we set $\widetilde{g}(t,y,z):=g(t,y+D_{t},z)$ for all
$(\omega,t,y,z)\in\Omega\times[0,T]\times\mathbb{R}\times\mathbb{R}^{n}$. To
simplify the presentation, we find it convenient to further transform the BSDE
(2.3) by setting $\widetilde{Y}:=Y-D,\,\widetilde{\eta}:=\eta-D_{T}$ and
$\widetilde{Z}=Z$. Then we obtain the transformed BSDE with modified data
$(\widetilde{g},\widetilde{\eta},0)$

$\displaystyle\widetilde{Y}_{t}=\widetilde{\eta}+\int_{t}^{T}\widetilde{g}(u,\widetilde{Y}_{u},\widetilde{Z}_{u})\,dQ_{u}-\int_{t}^{T}\widetilde{Z}_{u}^{\ast}\,dM_{u}.$
(4.1)

It is easy to see that, under Assumption 2.3, the random variable
$\widetilde{\eta}$ belongs to $L^{2}({\mathcal{F}}_{T})$, the modified
generator $\widetilde{g}$ is uniformly $m$-Lipschitz continuous with a
constant $\widehat{L}$ and the process $\widetilde{g}(\cdot,0,0)$ belongs to
${\mathcal{H}}^{2}(Q)$. Consistently with Definition 2.4 and Remark 2.3, we
now search for a solution $(\widetilde{Y},\widetilde{Z})\in\mathbb{H}^{2}$ to
the BSDE (4.1) (in particular, $\widetilde{Y}$ should be an RCLL process). It
is readily seen that the following lemma is valid.

######  Lemma 4.1.

Under Assumption 2.3, the existence and uniqueness of a solution
$(\widetilde{Y},\widetilde{Z})$ to the transformed BSDE (4.1) is equivalent to
the existence and uniqueness of a solution $(Y,Z)$ to the BSDE (2.2) and the
respective solutions are related to each other through the mapping
$(Y,Z)=(\widetilde{Y}+D,\widetilde{Z})$.

We are now ready to prove the existence and uniqueness theorem for solutions
to the BSDE (2.2) with generator satisfying the uniform $m$-Lipschitz
condition.

######  Theorem 4.1.

If Assumptions 2.1–2.3 are valid, then there exists a solution
$(Y,Z)\in\mathbb{H}^{2}$ to the BSDE (2.2) such that the process $Y-D$ belongs
to ${\mathcal{S}}^{2}$. Moreover, the uniqueness of a solution $(Y,Z)$ holds
in the space $\mathbb{H}^{2}$ and the uniqueness of $(Y-D,Z)$ holds in
$\mathbb{S}^{2}$.

Proof.  It is clear that both statements about uniqueness are immediate
consequences of inequality (3.2) in Proposition 3.1 since if $(Y,Z)$ and
$(Y^{\prime},Z^{\prime})$ are two solutions to the BSDE (2.2), then we have
$g^{1}=g^{2},\,D^{1}=D^{2}$ and $\eta^{1}=\eta^{2}$ so that (3.2) yields

$\displaystyle\|Y-Y^{\prime}\|_{{\mathcal{S}}^{2}}^{2}+\|Y-Y^{\prime}\|_{{\mathcal{H}}^{2}(Q)}^{2}+\|Z-Z^{\prime}\|_{{\mathcal{L}}^{2}(M)}^{2}=0$

where we observe that $Y-Y^{\prime}=(Y-D)-(Y^{\prime}-D)$. Hence it is enough
to prove the first assertion. In view of Lemma 4.1 and since $D$ belongs to
${\mathcal{H}}^{2}(Q)$, it would be possible to assume, without loss of
generality, that $D=0$.

In the first step, for a given pair $(w,v)\in\mathbb{S}^{2}_{\beta}$, we
examine the following BSDE

$\left\\{\begin{array}[c]{ll}dY_{t}=-g_{t}\,dQ_{t}+Z_{t}^{\ast}\,dM_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ Y_{T}=\eta,\end{array}\right.$ (4.2)

where $g_{t}:=g(t,w_{t},v_{t})$. To establish the existence of a unique
solution to (4.2), it suffices to apply the postulated martingale
representation property of $M$ to the square-integrable
$\mathbb{F}$-martingale
$N_{t}:={\mathbb{E}}\left[\eta+\int_{0}^{T}g_{u}\,dQ_{u}\big{|}\mathcal{F}_{t}\right]$.
Due to Assumption 2.2, there exists a process $Z\in{\mathcal{L}}^{2}(M)$ such
that, for all $t\in[0,T]$,

$\displaystyle
N_{t}={\mathbb{E}}\bigg{[}\eta+\int_{0}^{T}g_{u}\,dQ_{u}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}$

and the uniqueness of $Z$ in $\|\cdot\|_{{\mathcal{L}}^{2}(M)}$ holds. We
define the RCLL, $\mathbb{F}$-adapted process $Y$ by setting

$\displaystyle
Y_{t}:={\mathbb{E}}\bigg{[}\eta+\int_{t}^{T}g_{u}\,dQ_{u}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}=N_{t}-\int_{0}^{t}g_{u}\,dQ_{u}=\int_{0}^{t}Z^{*}_{u}\,dM_{u}-\int_{0}^{t}g_{u}\,dQ_{u}.$

Then we have, for all $t\in[0,T]$,

$\displaystyle
Y_{t}=\eta+\int_{t}^{T}g_{u}\,dQ_{u}-\int_{t}^{T}Z_{u}^{\ast}\,dM_{u}.$

It is now easy to check that the pair $(Y,Z)$ belongs to
$\mathbb{S}^{2}_{\beta}$ for any $\beta>0$ and is a solution to the BSDE
(4.2). Furthermore, the uniqueness of a solution to (4.2) in
$\mathbb{H}^{2}_{\beta}$ (and in $\mathbb{S}^{2}_{\beta}$) also follows from
inequality (3.2) in Proposition 3.1.

In the second step, for any $(w,v)\in\mathbb{S}^{2}_{\beta}$, we denote by
$(Y^{w,v},Z^{w,v})\in\mathbb{S}^{2}_{\beta}$ the unique solution to (4.2) with
$g_{t}:=g(t,w_{t},v_{t})$ and we define the mapping
$\Psi:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ as follows: for
any $(w,v)\in\mathbb{S}^{2}_{\beta}$, we set $\Psi(w,v)=(Y^{w,v},Z^{w,v})$. We
claim that $\Psi$ is a contraction mapping on the Banach space
$\mathbb{S}^{2}_{\beta}$, provided that $\beta$ is sufficiently large. To this
end, we proceed as follows: for $l=1,2$, we assume that $(Y^{l},Z^{l})$ is a
solution to BSDE (4.2) with generator $g^{l}_{t}:=g(t,w^{l}_{t},v^{l}_{t})$
where $(w^{l},v^{l})\in\mathbb{S}^{2}_{\beta}$, and we show that for an
arbitrary $\beta>0$

$\|y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
1158\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}$
(4.3)

where $y:=Y^{1}-Y^{2},\,z:=Z^{1}-Z^{2},\,w:=w^{1}-w^{2}$ and $v:=v^{1}-v^{2}$.

Assuming temporarily that (4.3) is valid, it is immediate from (4.3) that
$\Psi:\mathbb{S}^{2}_{\beta}\rightarrow\mathbb{S}^{2}_{\beta}$ is a
contraction if we take $\beta$ sufficiently large. Then, by the Banach
theorem, there exists a unique fixed point $(Y,Z)\in\mathbb{S}^{2}_{\beta}$ of
$\Psi$. Furthermore, it is not hard to see that $Y\in{\mathcal{H}}^{2}(Q)$,
due to the boundedness of $Q$ (when $D\neq 0$, we also have that
$Y\in{\mathcal{H}}^{2}(Q)$ since $D\in{\mathcal{H}}^{2}(Q)$). Thus the above
fixed point $(Y,Z)$ is a solution to BSDE (4.1) in $\mathbb{S}^{2}$ and thus
also in $\mathbb{H}^{2}$.

It is now clear that to complete the proof, it suffices to establish (4.3). By
taking the difference of solutions to BSDEs with generator
$g^{l}_{t}:=g(t,w^{l}_{t},v^{l}_{t})$ for $l=1,2$, we obtain

$\left\\{\begin{array}[c]{ll}dy_{t}=-\delta_{t}\,dQ_{t}+z^{*}_{t}\,dM_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=0,\end{array}\right.$

where $\delta_{t}:=g(t,w^{1}_{t},v^{1}_{t})-g(t,w^{2}_{t},v^{2}_{t})$. From
the first step, it is clear that for an arbitrary $\beta>0$

${\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y_{t}^{2}\big{)}\bigg{]}=\|y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}\leq\|Y^{1}\|^{2}_{{\mathcal{S}}^{2}_{\beta}}+\|Y^{2}\|^{2}_{{\mathcal{S}}^{2}_{\beta}}<\infty$
(4.4)

since $Y^{1}$ and $Y^{2}$ belong to ${\mathcal{S}}^{2}$ and thus also to
${\mathcal{S}}^{2}_{\beta}$. The Itô integration by parts formula gives

$\displaystyle d(e^{\beta Q_{t}}y^{2}_{t})=y^{2}_{t}\,de^{\beta
Q_{t}}+e^{\beta Q_{t}}\,dy^{2}_{t}=y^{2}_{t}\beta e^{\beta
Q_{t}}\,dQ_{t}+e^{\beta Q_{t}}\,dy^{2}_{t}$

where

$\displaystyle
dy^{2}_{t}=2y_{t-}\,dy_{t}+d[y]_{t}=2y_{t-}\,dy_{t}+dN_{t}+d\langle
y\rangle_{t}$

and $N:=[y]-\langle y\rangle$. Consequently,

$\begin{split}d(e^{\beta Q_{t}}y^{2}_{t})&=e^{\beta Q_{t}}\big{(}\beta
y^{2}_{t}\,dQ_{t}-2y_{t-}\delta_{t}\,dQ_{t}+2y_{t-}z^{*}_{t}\,dM_{t}+d[y]_{t}\big{)}\\\
&=e^{\beta Q_{t}}\big{(}\beta
y^{2}_{t}\,dQ_{t}-2y_{t-}\delta_{t}\,dQ_{t}+2y_{t-}z^{*}_{t}\,dM_{t}+dN_{t}+d\langle
y\rangle_{t}\big{)}.\end{split}$ (4.5)

To complete the derivation of (4.3), we first show that

$\|z\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\leq
2\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.$
(4.6)

After integrating (4.5) from $t$ to $T$, taking the expectation and using the
fact that $N$ is $\mathbb{F}$-martingale and $d\langle
y\rangle_{t}=\|m_{t}z_{t}\|^{2}\,dQ_{t}$, we obtain

$\displaystyle{\mathbb{E}}\bigg{[}e^{\beta
Q_{t}}y^{2}_{t}+\beta\,\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}+\int_{t}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}\bigg{]}$ $\displaystyle\leq
2\widehat{L}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|y_{u}|(|w_{u}|+\|m_{u}v_{u}\|)\,dQ_{u}\bigg{]}$ $\displaystyle\leq
2\lambda^{-1}\widehat{L}^{2}\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}\bigg{]}+\lambda\,{\mathbb{E}}\bigg{[}\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\bigg{]}.$

Let us take $\lambda=2\beta^{-1}\widehat{L}^{2}$. Then we get

$\displaystyle{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}\leq
2\beta^{-1}\widehat{L}^{2}\,{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}\bigg{]}+2\beta^{-1}\widehat{L}^{2}\,{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\bigg{]},$

which gives (4.6) since $Q$ is a nonnegative, nondecreasing process bounded by
$C_{Q}$.

In the remainder of the proof, we analyze the norm
$\|y\|_{{\mathcal{S}}^{2}_{\beta}}$ and we show that

$\|y\|^{2}_{{\mathcal{S}}^{2}_{\beta}}\leq
1156\beta^{-1}\widehat{L}^{2}(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.$
(4.7)

Using again (4.5) and noticing that the process $[y]$ is nondecreasing, we get
for any $\lambda>0$ (recall that
$\delta_{t}:=g(t,w^{1}_{t},v^{1}_{t})-g(t,w^{2}_{t},v^{2}_{t})$ and $g$ is
uniformly $m$-Lipschitz continuous)

$\displaystyle e^{\beta Q_{t}}y^{2}_{t}+\beta\,\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}\leq 2\,\int_{t}^{T}e^{\beta
Q_{u}}y_{u}\delta_{u}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}$ $\displaystyle\leq
2\lambda^{-1}\widehat{L}^{2}\int_{t}^{T}e^{\beta
Q_{u}}y^{2}_{u}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u},$

which in turn yields, by taking $\lambda=2\beta^{-1}\widehat{L}^{2}$

$\begin{split}&e^{\beta Q_{t}}y^{2}_{t}\leq\lambda\int_{t}^{T}e^{\beta
Q_{u}}|w_{u}|^{2}\,dQ_{u}+\lambda\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\\\
&\leq\lambda(C_{Q}+1)\left(\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}|w_{t}|^{2}\big{)}+\int_{t}^{T}e^{\beta
Q_{u}}\|m_{u}v_{u}\|^{2}\,dQ_{u}\right)-2\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}.\end{split}$ (4.8)

Let us denote $\widetilde{y}_{u}=e^{\frac{\beta}{2}Q_{u}}y_{u}$ and
$\widetilde{M}_{t}:=\int_{0}^{t}e^{\frac{\beta}{2}Q_{u}}z^{\ast}_{u}\,dM_{u}$.
The Davis inequality (see, e.g., Theorem 10.24 in He et al. [14] or Theorem
11.5.5 in Cohen and Elliott [5] with $p=1$) gives

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\widetilde{y}_{u-}\,d\widetilde{M}_{u}\Big{|}\right]\leq
2\sqrt{6}\,{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}\widetilde{y}_{u-}^{2}\,d[\widetilde{M}]_{u}\bigg{)}^{1/2}\right]\leq
6\,{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}\widetilde{y}_{u-}^{2}\,d[\widetilde{M}]_{u}\bigg{)}^{1/2}\right].$

Consequently, using also the elementary inequalities

$\displaystyle{\mathbb{E}}\left[\bigg{(}\int_{0}^{T}\widetilde{y}_{u-}^{2}\,d[\widetilde{M}]_{u}\bigg{)}^{1/2}\right]$
$\displaystyle\leq{\mathbb{E}}\bigg{[}[\widetilde{M}]^{\frac{1}{2}}_{T}\sup_{t\in[0,T]}|\widetilde{y}_{t}|\bigg{]}\leq\frac{1}{48}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\widetilde{y}^{2}_{t}\bigg{]}+12\,{\mathbb{E}}\big{(}[\widetilde{M}]_{T}\big{)}$
$\displaystyle=\frac{1}{48}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\widetilde{y}^{2}_{t}\bigg{]}+12\,{\mathbb{E}}\big{(}\langle\widetilde{M}\rangle_{T}\big{)},$

we obtain

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}\widetilde{y}_{u-}\,d\widetilde{M}_{u}\Big{|}\right]\leq\frac{1}{8}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\widetilde{y}^{2}_{t}\bigg{]}+72\,{\mathbb{E}}\big{(}\langle\widetilde{M}\rangle_{T}\big{)},$

which can be rewritten more explicitly as follows

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\Big{|}\right]\leq\frac{1}{8}\,{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y^{2}_{t}\big{)}\Big{]}+72\,{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}.$

Therefore, using also already proven inequality (4.6), we find that

$\displaystyle{\mathbb{E}}\left[\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\Big{|}\right]\leq\frac{1}{4}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y^{2}_{t}\big{)}\bigg{]}+144\,{\mathbb{E}}\bigg{[}\int_{0}^{T}e^{\beta
Q_{u}}\,\|m_{u}z_{u}\|^{2}\,dQ_{u}\bigg{]}$
$\displaystyle\leq\frac{1}{4}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y^{2}_{t}\big{)}\bigg{]}+144\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}.$

By combining (4.8) with the inequality above, we deduce that

$\displaystyle{\mathbb{E}}\Big{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y^{2}_{t}\big{)}\Big{]}$
$\displaystyle\leq\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}+2\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{|}\int_{t}^{T}e^{\beta
Q_{u}}y_{u-}z^{*}_{u}\,dM_{u}\Big{|}\bigg{]}$ $\displaystyle\leq
289\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)}+\frac{1}{2}\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y^{2}_{t}\big{)}\bigg{]}$

and thus, in view of the finiteness of the norm
$\|y\|_{{\mathcal{S}}^{2}_{\beta}}$ (see (4.4)), we obtain

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{(}e^{\beta
Q_{t}}y_{t}^{2}\big{)}\bigg{]}\leq
578\lambda(C_{Q}+1)\big{(}\|w\|_{{\mathcal{S}}^{2}_{\beta}}^{2}+\|v\|^{2}_{{\mathcal{L}}^{2}_{\beta}(M)}\big{)},$

which shows that (4.7) is satisfied (recall that
$\lambda=2\beta^{-1}\widehat{L}^{2}$). Since inequalities (4.6) and (4.7) are
now confirmed, we conclude that the desired inequality (4.3) is valid as well
and thus the proof of the theorem is now complete. $\Box$

##  5 Solution to the Linear BSDE

When examining the special case of the linear BSDE, we work under two
additional postulates, which will allow us to find an explicit representation
for the component $Y$ of a unique solution $(Y,Z)$. The first one concerns the
structure of the predictable covariation process $\langle M\rangle$.

######  Assumption 5.1.

The matrix $m_{t}$ in representation (2.1) of the predictable covariation
process $\langle M\rangle$ is invertible for all $t\in[0,T]$.

Assumption 5.1 allows us to define the auxiliary function
$\widehat{g}(t,y,z):=g(t,y,m^{-1}_{t}z)$. It is readily seen that if $g$ is an
arbitrary uniformly $m$-Lipschitz continuous generator, then the function
$\widehat{g}$ is uniformly Lipschitz continuous since we have

$|\widehat{g}(t,y_{1},z_{1})-\widehat{g}(t,y_{2},z_{2})|\leq\widehat{L}\big{(}|y_{1}-y_{2}|+\|z_{1}-z_{2}\|\big{)}.$

We also impose a specific structure on the $n$-dimensional martingale $M$.

######  Assumption 5.2.

We assume that $M=(M^{1},\dots,M^{n})^{*}=(M^{c},M^{d})^{\ast}$ where the
process $M^{c}:=(M^{1},\dots,M^{k})^{*}$ (respectively,
$M^{d}=(M^{k+1},\dots,M^{n})^{*}$) is an $\mathbb{R}^{k}$-valued, continuous
martingale (respectively, an $\mathbb{R}^{n-k}$-valued, purely discontinuous
martingale) for some $k\in\\{0,1,\dots,n\\}$.

######  Remark 5.1.

We stress that Assumption 5.2 is not directly concerned with the dynamics of
asset prices, but rather with modeling of underlying sources for various kinds
of financial risks. Therefore, it is not restrictive when dealing with
practical applications of BSDEs driven by an RCLL martingale to problems
arising in financial mathematics (see, in particular, Dumitrescu et al. [8] or
Kim et al. [19, 20] for recent studies of American and game options in
nonlinear markets).

Let us denote $n_{t}:=m_{t}^{-1}$ so that $n_{t}^{-1}=m_{t}=m_{t}^{\ast}$.
Under Assumption 5.2, we have that $\langle M^{i},M^{j}\rangle=0$ for all
$i,j$ such that $i\leq k$ and $j\geq k+1$. Therefore, we may and do assume
that $m$ is a symmetric block matrix and thus the notation
$n^{c}_{t}=(m^{c}_{t})^{-1}$ and $n^{d}_{t}=(m^{d}_{t})^{-1}$ where
$m^{c}_{t}$ and $m^{d}_{t}$ are symmetric random matrices of dimensions $k$
and $n-k$, respectively, can be easily justified. By the same token, for any
$z=(z_{1},\dots,z_{n})^{*}\in\mathbb{R}^{n}$ we write $z=(z^{c},z^{d})^{*}$
where $z^{c}=(z_{1},\dots,z_{k})$ and $z^{d}=(z_{k+1},\dots,z_{n})$. An
analogous notational convention can be applied to the process $Z$, so that we
may write $Z=(Z^{c},Z^{d})^{*}$ and to other stochastic processes of interest.

We now examine the linear BSDE

$\left\\{\begin{array}[c]{ll}dY_{t}=-\big{(}a_{t}Y_{t}+b_{t}+c^{*}_{t}m^{c}_{t}Z^{c}_{t}+d^{*}_{t}m^{d}_{t}Z^{d}_{t}\big{)}\,dQ_{t}+Z_{t}^{\ast}\,dM_{t}+dD_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ Y_{T}=\eta,\end{array}\right.$ (5.1)

and we work under the following variant of Assumption 2.3.

######  Assumption 5.3.

The triplet $(g,\eta,D)$ is such that: (i) the generator $g$ equals
$g(t,y,z)=a_{t}y+b_{t}+c^{*}_{t}m^{c}_{t}z^{c}+d^{*}_{t}m^{d}_{t}z^{d}$ where
the real-valued processes $a,c$ and $d$ are bounded and $b$ belongs to
${\mathcal{H}}^{2}(Q)$. (ii) the random variable $\eta$ belongs to
$L^{2}({\mathcal{F}}_{T})$, (iii) $D$ is an RCLL process of finite variation
and belongs to ${\mathcal{H}}^{2}(Q)$; furthermore, $|D|_{T}\in
L^{2}({\mathcal{F}}_{T})$ where $|D|=(|D|_{t},\,t\in[0,T])$ is the total
variation process of $D$.

######  Remark 5.2.

Notice that the total variation of $D$ satisfies

$\displaystyle|D|_{T}\geq\sum_{u\leq t}|\Delta D_{u}|\geq\bigg{(}\sum_{u\leq
t}(\Delta D_{u})^{2}\bigg{)}^{1/2}.$

Hence we deduce from part (iii) in Assumption 5.3 that
${\mathbb{E}}\big{(}\sum_{u\leq t}(\Delta D_{u})^{2}\big{)}<\infty$.

Since the processes $a,c,d$ are assumed to be bounded, the generator
$g(t,y,z)=a_{t}y+b_{t}+c^{*}_{t}m^{c}_{t}z^{c}+d^{*}_{t}m^{d}_{t}z^{d}$ is
uniformly $m$-Lipschitz continuous and the process $g(\cdot,0,0)=c$ belongs to
${\mathcal{H}}^{2}(Q)$. Therefore, we may apply Theorem 4.1 to ensure that the
BSDE (5.1) has a unique solution $(Y,Z)\in{\mathcal{S}}^{2}$ provided that
Assumptions 2.1-2.2 and 5.1-5.3 are satisfied. Our goal is to find an explicit
representation for the process $Y$, which is convenient to establish the
comparison property.

Before stating the next result, we need to introduce some notation. For an
arbitrary $\mathbb{R}^{n-k}$-valued process $d$, which is assumed to be
bounded and $\mathbb{F}$-predictable, we define the RCLL, $\mathbb{F}$-adapted
process $q$ by

$q_{t}:=\mathcal{E}_{t}(\widehat{Q})\mathcal{E}_{t}(\widehat{M})=\mathcal{E}_{t}(\widehat{Q}+\widehat{M})$
(5.2)

where $\widehat{Q}$ is a continuous, bounded process of finite variation given
by $\widehat{Q}_{t}:=\int_{0}^{t}a_{u}\,dQ_{u}$ and $\widehat{M}$ is an
$\mathbb{F}$-martingale defined by

$\widehat{M}_{t}:=\int_{0}^{t}c^{*}_{u}n^{c}_{u}\,dM^{c}_{u}+\int_{0}^{t}d^{\ast}_{u}n^{d}_{u}\,dM^{d}_{u}=\widehat{M}^{c}_{t}+\widehat{M}^{d}_{t}.$
(5.3)

Here $\mathcal{E}(\widehat{M})$ denotes the Doléans exponential given by the
following expression

$\displaystyle\mathcal{E}_{t}(\widehat{M})=\exp\Big{(}\widehat{M}_{t}-\frac{1}{2}\langle\widehat{M}^{c}\rangle_{t}\Big{)}\prod_{0<u\leq
t}\big{(}1+\Delta\widehat{M}_{u}\big{)}e^{-\Delta\widehat{M}_{u}}.$

For brevity, we denote

$\Psi^{q,D}_{t}:=\int_{0}^{t}q_{u-}\,dD_{u}+\langle q,D\rangle_{t}.$ (5.4)

We are ready to derive an explicit representation for the process $Y$ in the
unique solution to the linear BSDE.

######  Proposition 5.1.

If Assumptions 2.1-2.2 and 5.1-5.3 hold, then the linear BSDE (5.1) has a
unique solution $(Y,Z)\in{\mathcal{S}}^{2}$ and the process $Y$ satisfies

$q_{t}Y_{t}={\mathbb{E}}\bigg{[}q_{T}\eta+\int_{t}^{T}q_{u}b_{u}\,dQ_{u}-\big{(}\Psi^{q,D}_{T}-\Psi^{q,D}_{t}\big{)}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}$
(5.5)

where $q$ and $\Psi^{q,D}$ are given by (5.2) and (5.4), respectively.

Proof. Since $\widehat{Q}$ is an $\mathbb{F}$-adapted, bounded, continuous
process of finite variation and $\widehat{M}$ is an $\mathbb{F}$-martingale
given by (5.3), it is obvious that $[\widehat{Q},\widehat{M}]=0$. Recall that
$\mathcal{E}(\widehat{M})$ is given by

$\displaystyle\mathcal{E}_{t}(\widehat{M})=\exp\Big{(}\widehat{M}_{t}-\frac{1}{2}\langle\widehat{M}^{c}\rangle_{t}\Big{)}\prod_{0<u\leq
t}\big{(}1+\Delta\widehat{M}_{u}\big{)}e^{-\Delta\widehat{M}_{u}}$

where $\widehat{M}^{c}_{t}=\int_{0}^{t}c^{*}_{u}n^{c}_{u}\,dM^{c}_{u}$ and
$\Delta\widehat{M}_{u}:=\widehat{M}_{u}-\widehat{M}_{u-}=\widehat{M}^{d}_{u}-\widehat{M}^{d}_{u-}=d^{*}_{u}n^{d}_{u}\Delta
M^{d}_{u}$. Simple computations show that $q$ can be represented as follows

$\displaystyle
q_{t}=\exp\left(\int_{0}^{t}a_{u}\,dQ_{u}\right)\exp\left(\widehat{M}^{c}_{t}-\frac{1}{2}\int_{0}^{t}\|c_{u}\|^{2}\,dQ_{u}\right)\exp\big{(}\widehat{M}^{d}_{t}\big{)}\prod_{0<u\leq
t}\big{(}1+\Delta\widehat{M}^{d}_{u}\big{)}e^{-\Delta\widehat{M}^{d}_{u}}.$

In view of (5.2) and (5.3), we have that

$\displaystyle
dq_{t}=q_{t-}\,d(\widehat{Q}_{t}+\widehat{M}_{t})=q_{t-}\,d\widehat{Q}_{t}+q_{t-}\,d\widehat{M}_{t}=q_{t-}\big{(}a_{t}\,dQ_{t}+c^{*}_{t}n^{c}_{t}\,dM^{c}_{t}+d^{\ast}_{t}n^{d}_{t}\,dM^{d}_{t}\big{)}$

and thus, using (5.1), we obtain

$d\langle
q,Y\rangle_{t}=q_{t}\big{(}c^{*}_{t}m^{c}_{t}Z^{c}_{t}+d^{\ast}_{t}m^{d}_{t}Z^{d}_{t}\big{)}\,dQ_{t}+\langle
q,D\rangle_{t}$

where we denote $Z=(Z^{c},Z^{d})^{*}$. Since $D$ is an RCLL process of finite
variation, and we have

$[q,D]_{t}=\sum_{u\leq t}\Delta q_{u}\Delta D_{u}$

where $\Delta q_{t}=q_{t}-q_{t-}$ and $\Delta D_{t}=D_{t}-D_{t-}$. Using the
Itô integration by parts formula, we obtain

$\displaystyle d(q_{t}Y_{t})$
$\displaystyle=q_{t-}\,dY_{t}+Y_{t-}\,dq_{t}+d[q,Y]_{t}$
$\displaystyle=-q_{t}\big{(}a_{t}Y_{t}+b_{t}+c^{*}_{t}m^{c}_{t}Z^{c}_{t}+d^{*}_{t}m^{d}_{t}Z^{d}_{t}\big{)}\,dQ_{t}+q_{t-}Z_{t}^{\ast}\,dM_{t}+q_{t-}\,dD_{t}$
$\displaystyle\quad\mbox{}+q_{t}Y_{t}a_{t}\,dQ_{t}+q_{t-}Y_{t-}\big{(}c^{*}_{t}n^{c}_{t}\,dM^{c}_{t}+d^{\ast}_{t}n^{d}_{t}\,dM^{d}_{t}\big{)}+d\langle
q,Y\rangle_{t}+d[q,Y]_{t}-d\langle q,Y\rangle_{t}$
$\displaystyle=d\widetilde{M}_{t}-q_{t}\big{(}b_{t}+c^{*}_{t}m^{c}_{t}Z^{c}_{t}+d^{*}_{t}m^{d}_{t}Z^{d}_{t}\big{)}\,dQ_{t}+q_{t}\big{(}c^{*}_{t}m^{c}_{t}Z^{c}_{t}+d^{\ast}_{t}m^{d}_{t}Z^{d}_{t}\big{)}\,dQ_{t}+q_{t-}\,dD_{t}+d\langle
q,D\rangle_{t}$
$\displaystyle=d\widetilde{M}_{t}-q_{t}b_{t}\,dQ_{t}+d\Psi^{q,D}_{t}$

where the RCLL process $\widetilde{M}$ is defined by

$d\widetilde{M}_{t}=q_{t-}\big{(}Z_{t}^{\ast}\,dM_{t}+Y_{t-}c^{*}_{t}n^{c}_{t}\,dM^{c}_{t}+Y_{t-}d^{\ast}_{t}n^{d}_{t}\,dM^{d}_{t}\big{)}+d[q,Y]_{t}-d\langle
q,Y\rangle_{t}$

where $\widetilde{M}_{0}:=q_{0}Y_{0}=Y_{0}$ and the process $N:=[q,Y]-\langle
q,Y\rangle$ is an $\mathbb{F}$-local martingale with $N_{0}=0$. We also have
that

$\displaystyle\widetilde{M}_{T}-\widetilde{M}_{t}=q_{T}\eta-
q_{t}Y_{t}+\int_{t}^{T}q_{u}b_{u}\,dQ_{u}-(\Psi^{q,D}_{T}-\Psi^{q,D}_{t})$

and thus, in view of Lemma 5.1, the process $\widetilde{M}$ is an
$\mathbb{F}$-martingale. Hence, after taking the conditional expectation with
respect to $\mathcal{F}_{t}$, we obtain (5.5). $\Box$

The following lemma is required for the proof of Proposition 5.1 and Theorem
6.1.

######  Lemma 5.1.

If Assumptions 2.1-2.2 and 5.1-5.3 are satisfied, then the process
$\widetilde{M}$ given by

$\widetilde{M}_{t}=q_{t}Y_{t}-\Psi^{q,D}_{t}+\int_{0}^{t}q_{u}b_{u}\,dQ_{u},$
(5.6)

is an $\mathbb{F}$-martingale.

Proof. We claim that

${\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}|\widetilde{M}_{t}|\bigg{]}<\infty,$
(5.7)

which is known to imply that $\widetilde{M}$ is a uniformly integrable
$\mathbb{F}$-martingale. Our goal is thus to establish (5.7). For this
purpose, we start by examining the process $q$. Since $c$ and $d$ are bounded,
we have, for some constant $C$,

$\langle\widehat{M}\rangle_{T}=\int_{0}^{T}\big{(}|c_{t}|^{2}+|d_{t}|^{2}\big{)}\,dQ_{t}\leq
C<\infty.$ (5.8)

We will deduce from (5.2) that

$\|q\|^{2}_{{\mathcal{S}}^{2}}={\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}q_{t}^{2}\bigg{]}<\infty.$
(5.9)

To this end, we first observe that (5.2) gives
$q_{t}=\mathcal{E}_{t}(\widehat{Q})\mathcal{E}_{t}(\widehat{M})$ where the
process $\mathcal{E}(\widehat{Q})$ is bounded since $a$ and $Q$ are bounded.
To establish (5.9), it thus suffices to show that

$\|\mathcal{E}(\widehat{M})\|^{2}_{{\mathcal{S}}^{2}}={\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\mathcal{E}_{t}^{2}(\widehat{M})\bigg{]}<\infty.$
(5.10)

Observe that

$\displaystyle
d\mathcal{E}_{t}(\widehat{M})=\mathcal{E}_{t-}(\widehat{M})\,d\widehat{M}_{t}=\mathcal{E}_{t-}(\widehat{M})\,d\widehat{M}^{c}_{t}+\mathcal{E}_{t-}(\widehat{M})\,d\widehat{M}^{d}_{t}$

and thus

$\displaystyle d[\mathcal{E}(\widehat{M})]_{t}$
$\displaystyle=\mathcal{E}_{t-}^{2}(\widehat{M})\,d\langle\widehat{M}^{c}\rangle_{t}+\mathcal{E}_{t-}^{2}(\widehat{M})\,d[\widehat{M}^{d}]_{t}$
$\displaystyle=\mathcal{E}_{t-}^{2}(\widehat{M})\big{(}d\langle\widehat{M}^{c}\rangle_{t}+d\langle\widehat{M}^{d}\rangle_{t}\big{)}+\mathcal{E}_{t-}^{2}(\widehat{M})\big{(}d[\widehat{M}^{d}]_{t}-d\langle\widehat{M}^{d}\rangle_{t}\big{)},$

which in turn yields

$\displaystyle d\mathcal{E}_{t}^{2}(\widehat{M})$
$\displaystyle=2\mathcal{E}_{t-}(\widehat{M})\,d\mathcal{E}_{t}(\widehat{M})+d[\mathcal{E}(\widehat{M})]_{t}$
$\displaystyle=\mathcal{E}_{t-}^{2}(\widehat{M})\big{(}d\langle\widehat{M}^{c}\rangle_{t}+d\langle\widehat{M}^{d}\rangle_{t}\big{)}+\mathcal{E}_{t-}^{2}(\widehat{M})\big{(}2\,d\widehat{M}_{t}+d[\widehat{M}^{d}]_{t}-d\langle\widehat{M}^{d}\rangle_{t}\big{)}.$

Recalling that $\mathcal{E}_{0}(\widehat{M})=1$, we obtain
$\mathcal{E}_{t}^{2}(\widehat{M})=\gamma_{t}\exp\big{(}\langle\widehat{M}^{c}\rangle_{t}+\langle\widehat{M}^{d}\rangle_{t}\big{)}$
where $\gamma_{0}=1$ and

$\displaystyle
d\gamma_{t}=\gamma_{t-}\Big{(}2\,d\widehat{M}_{t}+d[\widehat{M}^{d}]_{t}-d\langle\widehat{M}^{d}\rangle_{t}\Big{)}.$

From the last two equations, it follows that $\gamma$ is a non-negative local
martingale and hence a supermartingale, so that
${\mathbb{E}}[\gamma_{t}]\leq{\mathbb{E}}[\gamma_{0}]=1$ for all $t\in[0,T]$.
It is clear from (5.8) that there exists a constant $C$ such that, for all
$t\in[0,T]$

$\displaystyle\exp\big{(}\langle\widehat{M}^{c}\rangle_{t}+\langle\widehat{M}^{d}\rangle_{t}\big{)}\leq
C.$

Consequently, it follows that $\mathcal{E}_{t}^{2}(\widehat{M})\leq
C\gamma_{t}$ for all $t\in[0,T]$. Since $\mathcal{E}^{2}(\widehat{M})$ and
$\gamma$ are RCLL processes, we deduce that
$\mathcal{E}_{\tau}^{2}(\widehat{M})\leq C\gamma_{\tau}$ for all
$\tau\in\mathcal{T}$ where $\mathcal{T}$ is the set of all stopping times
taking values in $[0,T]$. By applying the optional stopping theorem to
$\gamma$, we get

$\displaystyle{\mathbb{E}}\Big{[}\mathcal{E}_{\tau}^{2}(\widehat{M})\Big{]}\leq
C\,{\mathbb{E}}\big{[}\gamma_{\tau}\big{]}\leq C,\text{ for all
}\tau\in\mathcal{T}.$

Hence the family $\\{\mathcal{E}_{\tau}(\widehat{M})\\}_{\tau\in\mathcal{T}}$
is uniformly integrable, which implies that $\mathcal{E}(\widehat{M})$ is a
nonnegative RCLL martingale. From Doob’s inequality, we obtain

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\mathcal{E}_{t}^{2}(\widehat{M})\bigg{]}\leq
4\,{\mathbb{E}}\big{[}\mathcal{E}_{T}^{2}(\widehat{M})\big{]}\leq 4C,$

which shows that (5.10) is valid and thus (5.9) holds as well.

We are now ready to analyze the right-hand side of (5.6). It is clear that

$\displaystyle\big{|}q_{t}Y_{t}-\Psi^{q,D}_{t}\big{|}\leq|q_{t}||Y_{t}-D_{t}|+|q_{t}||D_{t}|+(|D_{0}|+|D_{T}|)\sup_{t\in[0,T]}q_{t}+\sup_{t\in[0,T]}|\langle
q,D\rangle_{t}|.$

Since $D$ is assumed to be a process of finite variation with the total
variation process $|D|$ such that ${\mathbb{E}}(|D|^{2}_{T})<\infty$, we have
$\sup_{\,t\in[0,T]}D^{2}_{t}\leq 2(D^{2}_{0}+|D|^{2}_{T})<\infty$ and thus $D$
belongs to ${\mathcal{S}}^{2}$. Of course, the process $Y-D$ is in
${\mathcal{S}}^{2}$ as well (see Theorem 4.1). Using the Kunita-Watanabe
inequality (see, for instance, Theorem 8.3 in He et al. [14]), we obtain, for
all $t\in[0,T]$,

$\displaystyle|\langle q,D\rangle_{t}|\leq(\langle q\rangle_{t}\langle
D\rangle_{t})^{1/2}\leq 2(\langle q\rangle_{t}+\langle D\rangle_{t})\leq
2(\langle q\rangle_{T}+\langle D\rangle_{T})$

and thus

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}|\langle
q,D\rangle_{t}|\bigg{]}\leq 2\big{(}{\mathbb{E}}(\langle
q\rangle_{T})+{\mathbb{E}}(\langle D\rangle_{T})\big{)}.$

Since $D$ is a process of finite variation, we have $[D]_{t}=\sum_{u\leq
t}(\Delta D_{u})^{2}$ and, from part (iii) in Assumption 5.3, we infer that
${\mathbb{E}}([D]_{T})<\infty$ (see Remark 5.2), which in turn implies that
${\mathbb{E}}(\langle D\rangle_{T})<\infty$. Similarly, since

$\displaystyle
dq_{t}=d(\mathcal{E}_{t}(\widehat{Q})\mathcal{E}_{t}(\widehat{M}))=\mathcal{E}_{t}(\widehat{Q})\,d\mathcal{E}_{t}(\widehat{M})+\mathcal{E}_{t-}(\widehat{M})\,d\mathcal{E}_{t}(\widehat{Q})$

and
$\langle\mathcal{E}(\widehat{Q})\rangle=\langle\mathcal{E}(\widehat{Q}),\mathcal{E}(\widehat{M})\rangle=0$,
using the boundedness of the processes $\mathcal{E}(\widehat{Q})$ and
$\langle\widehat{M}\rangle$ (see (5.8)), we obtain

$\displaystyle{\mathbb{E}}(\langle
q\rangle_{T})={\mathbb{E}}\bigg{[}\int_{0}^{T}\mathcal{E}^{2}_{t}(\widehat{Q})\mathcal{E}^{2}_{t-}(\widehat{M})\,d\langle\widehat{M}\rangle_{t}\bigg{]}\leq
C\|\mathcal{E}(\widehat{M})\|^{2}_{{\mathcal{S}}^{2}}.$

We conclude that

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\big{|}q_{t}Y_{t}-\Psi^{q,D}_{t}\big{|}\bigg{]}\leq
C\big{[}\|q\|_{{\mathcal{S}}^{2}}(\|y-D\|_{{\mathcal{S}}^{2}}+\|D\|_{{\mathcal{S}}^{2}})+{\mathbb{E}}(\langle
D\rangle_{T})+\|\mathcal{E}(\widehat{M})\|^{2}_{{\mathcal{S}}^{2}}\big{]}<\infty$

where we have also used (5.9) and (5.10).

Now let us study the integral $\int_{0}^{t}q_{u}b_{u}\,dQ_{u}$. Using the
boundedness of $Q$, we obtain the existence of constant $C$ which may very
line by line such that

$\displaystyle{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\Big{|}\int_{0}^{t}q_{u}b_{u}\,dQ_{u}\Big{|}\bigg{]}$
$\displaystyle\leq{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}\bigg{(}\int_{0}^{t}q^{2}_{u}b^{2}_{u}\,dQ_{u}\bigg{)}^{1/2}Q_{T}^{1/2}\bigg{]}\leq
C\,{\mathbb{E}}\bigg{[}\bigg{(}\int_{0}^{T}q^{2}_{u}b^{2}_{u}\,dQ_{u}\bigg{)}^{1/2}\bigg{]}$
$\displaystyle\leq
C\,{\mathbb{E}}\bigg{[}\sup_{t\in[0,T]}q_{t}\bigg{(}\int_{0}^{T}b^{2}_{u}\,dQ_{u}\bigg{)}^{1/2}\bigg{]}\leq
2C(\|q\|_{{\mathcal{S}}^{2}}+\|b\|_{{\mathcal{H}}^{2}(Q)})<\infty$

where we have also used (5.9) in the last inequality. We have thus shown that
(5.7) holds. $\Box$

Let us examine some consequences of Proposition 5.1. If we assume, in
addition, that $\Delta\widehat{M}_{t}=\Delta\widehat{M}^{d}_{t}>-1$ for all
$t\in[0,T]$, then the process $q$ given by (5.2) is strictly positive and thus
(5.5) yields the following representation for $Y$

$\displaystyle
Y_{t}={\mathbb{E}}\bigg{[}\Gamma_{t,T}\eta+\int_{t}^{T}\Gamma_{t,u}b_{u}\,dQ_{u}-\int_{t}^{T}\Gamma_{t,u-}\,dD_{u}-q_{t}^{-1}\big{(}\langle
q,D\rangle_{T}-\langle
q,D\rangle_{t}\big{)}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}$

where we denote $\Gamma_{t,u}=q_{t}^{-1}q_{u}$ for all $0\leq t\leq u\leq T$.

In particular, if the martingale $M$, and thus also the process $q$, are
continuous, we obtain the following corollary, which furnishes a minor
extension of the classical representation of a solution to a linear BSDE
driven by the Brownian motion (see, for instance, Proposition 2.2 in El Karoui
et al. [12]). Since $M$ is continuous, it is clear that $m=m^{c}$ (hence
$n=n^{c}$) and thus there is no need to introduce the processes $m^{d},n^{d}$
and $d$ in (5.1) and (5.3). Therefore, these equations reduce to

$\left\\{\begin{array}[c]{ll}dY_{t}=-\big{(}a_{t}Y_{t}+b_{t}+c^{*}_{t}m_{t}Z_{t}\big{)}\,dQ_{t}+Z_{t}^{\ast}\,dM_{t}+dD_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ Y_{T}=\eta,\end{array}\right.$ (5.11)

and

$\widehat{M}_{t}:=\int_{0}^{t}c^{*}_{u}n_{u}\,dM_{u}.$ (5.12)

######  Corollary 5.1.

Let Assumptions 2.1-2.2 and 5.1-5.3 hold so that the linear BSDE (5.11) has a
unique solution $(Y,Z)\in{\mathcal{S}}^{2}$. If the $\mathbb{F}$-martingale
$M$ is continuous, then $Y$ equals

$\displaystyle
Y_{t}={\mathbb{E}}\bigg{[}\Gamma_{t,T}\eta+\int_{t}^{T}\Gamma_{t,u}b_{u}\,dQ_{u}-\int_{t}^{T}\Gamma_{t,u}\,dD_{u}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}$

where $\Gamma_{t,u}=q_{t}^{-1}q_{u}$ with
$q_{t}=\mathcal{E}_{t}(\widehat{Q})\mathcal{E}_{t}(\widehat{M})$ where
$\widehat{Q}_{t}=\int_{0}^{t}a_{u}\,dQ_{u}$ and
$\widehat{M}_{t}=\int_{0}^{t}c^{*}_{u}n_{u}\,dM_{u}$.

##  6 Strict Comparison Theorem

An important feature of solutions to a BSDE is the so-called (strict)
comparison property (see the statement of Theorem 6.1), which underpins
several important applications of BSDEs to arbitrage-free pricing. Therefore,
our final goal is to establish the comparison theorem for the BSDE (3.1)
within the setup given by Assumptions 2.1, 2.2, 5.1 and 5.2. To prove the
comparison property of solutions to the BSDE (3.1) by the well-known method of
linearization, we define the following auxiliary processes

$a_{t}:=\frac{g^{1}(t,Y^{1}_{t},Z^{1}_{t})-g^{1}(t,Y^{2}_{t},Z^{1}_{t})}{Y^{1}_{t}-Y^{2}_{t}}\,\mathds{1}_{\\{Y^{1}_{t}\neq
Y^{2}_{t}\\}},$ $\displaystyle
c^{1}_{t}:=\frac{1}{\widetilde{Z}^{1}_{t,1}-\widetilde{Z}^{2}_{t,1}}\Big{[}\widehat{g}^{1}(t,Y^{2}_{t},\widetilde{Z}^{1}_{t,1},\widetilde{Z}^{1}_{t,2}\ldots,\ldots,\widetilde{Z}^{1}_{t,d})-\widehat{g}^{1}(t,Y^{2}_{t},\widetilde{Z}^{2}_{t,1},\widetilde{Z}^{1}_{t,2}\ldots,\ldots,\widetilde{Z}^{1}_{t,d})\Big{]}\mathds{1}_{\widetilde{D}_{1}},$

and, for every $j=2,3,\ldots,k$,

$\displaystyle c^{j}_{t}$
$\displaystyle:=\frac{1}{\widetilde{Z}^{1}_{t,j}-\widetilde{Z}^{2}_{t,j}}\Big{[}\widehat{g}^{1}(t,Y^{2}_{t},\widetilde{Z}^{2}_{t,1},\ldots,\widetilde{Z}^{2}_{t,j-1},\widetilde{Z}^{1}_{t,j},\widetilde{Z}^{1}_{t,j+1},\ldots,\widetilde{Z}^{1}_{t,d})$
$\displaystyle-\widehat{g}^{1}(t,Y^{2}_{t},\widetilde{Z}^{2}_{t,1},\ldots,\widetilde{Z}^{2}_{t,j-1},\widetilde{Z}^{2}_{t,j},\widetilde{Z}^{1}_{t,j+1},\ldots,\widetilde{Z}^{1}_{t,d})\Big{]}\mathds{1}_{\widetilde{D}_{j}}$

where $\widetilde{Z}_{t,j}^{i}:=(m_{t}Z_{t}^{i})_{j}$ for $i=1,2$ and
$\widetilde{D}_{j}:=\\{\widetilde{Z}^{1}_{t,j}\neq\widetilde{Z}^{2}_{t,j}\\}$
for $j=1,2,\dots,k$. For brevity, we denote by $c$ the $\mathbb{R}^{k}$-valued
process $c=(c^{1},\dots,c^{k})^{*}$. Recall that
$\widehat{g}(t,y,z):=g(t,y,m^{-1}_{t}z)$ and thus if $g$ is uniformly
$m$-Lipschitz continuous, then $\widehat{g}$ is uniformly Lipschitz
continuous.

The following result extends the comparison theorem established in [24] to the
case of BSDEs driven by RCLL martingales. The reader is also referred to
Theorem 3 in Dumitrescu et al. [7] and Theorem 3.2 in Peng and Xu [27] (see
also Theorem 3.3 in Peng and Xu [28]) for analogous, albeit less general,
results (see also Theorem 2.2 in Carbone et al. [4] for another instance of a
related comparison theorem for BSDEs). Recall that $\mu_{Q}$ denotes the
Doléans measure of the process $Q$ on the space $\Omega\times[0,T]$ endowed
with the product $\sigma$-algebra $\mathcal{F}_{T}\otimes\mathcal{B}([0,T])$
so that
$\mu_{Q}(A)={\mathbb{E}}\big{[}\int_{0}^{T}\mathds{1}_{A}(\omega,t)\,dQ_{t}(\omega)\big{]}$
for every $A\in\mathcal{F}_{T}\otimes\mathcal{B}([0,T])$.

######  Theorem 6.1.

Suppose that Assumptions 2.1, 2.2, 5.1 and 5.2 hold with an
$\mathbb{F}$-progressively measurable process $m$. Let Assumption 2.3 be
satisfied by the triplet $(g^{l},\eta^{l},D^{l})$ for $l=1,2$ and let
$(Y^{l},Z^{l})$ be the unique solution to the BSDE (3.1). Assume that: (i) the
processes $g^{l}(\cdot,\cdot,y,z),\,l=1,2$ are $\mathbb{F}$-predictable for
any fixed $(y,z)\in\mathbb{R}\times\mathbb{R}^{n}$, (ii) the process
$D:=D^{1}-D^{2}$ is RCLL, nonincreasing. (iii) the inequality
$\eta^{1}\geq\eta^{2}$ is valid,
(iv) there exists a bounded process $\zeta$ such that
$\Delta\widehat{M}_{t}=\Delta\widehat{M}^{d}_{t}>-1$ for every $t\in[0,T]$
where

$\widehat{M}_{t}:=\int_{0}^{t}c^{*}_{u}n^{c}_{u}\,dM^{c}_{u}+\int_{0}^{t}\zeta^{\ast}_{u}n^{d}_{u}\,dM^{d}_{u}=\widehat{M}^{c}_{t}+\widehat{M}^{d}_{t},$

and the following inequality holds

$b_{t}:=\delta_{t}-\zeta^{\ast}_{t}m^{d}_{t}(Z_{t}^{1,d}-Z_{t}^{2,d})\geq
0,\quad\mu_{Q}\mbox{\rm-a.e.,}$ (6.1)

where the process $\delta$ is given by

$\displaystyle\delta_{t}:=g^{1}(t,Y^{2}_{t},Z^{2}_{t,1},\ldots,Z^{2}_{t,k},Z^{1}_{t,k+1},Z^{1}_{t,k+2},\ldots,Z^{1}_{t,d})-g^{2}(t,Y^{2}_{t},Z^{2}_{t,1},\ldots,Z^{2}_{t,k},Z^{2}_{t,k+1},Z^{2}_{t,k+2},\ldots,Z^{2}_{t,d}).$

Then the following assertions are true:
(a) the comparison property is valid, that is, the inequality $Y^{1}_{t}\geq
Y^{2}_{t}$ holds for every $t\in[0,T]$,
(b) the strict comparison property holds, meaning that if
$Y^{1}_{0}=Y^{2}_{0}$, then $Y^{1}_{t}=Y^{2}_{t}$ for every $t\in[0,T]$ and
thus, in particular, $\eta^{1}=\eta^{2}$.

Proof. Since $(g^{l},\eta^{l},D^{l})$ satisfies Assumption 2.3 for $l=1,2$,
then BSDE (3.1) has the unique solution $(Y^{l},Z^{l})$ such that
$(Y^{l}-D^{l},Z^{l})\in\mathbb{S}^{2}$. We denote $g:=g^{1}-g^{2}$ and

$y:=Y^{1}-Y^{2},\ z:=Z^{1}-Z^{2},\ D:=D^{1}-D^{2},\ \eta:=\eta^{1}-\eta^{2}.$

It is straightforward to check that the pair $(y,z)$ satisfies
$(y-D,z)\in\mathbb{S}^{2}$ and solves the following linear BSDE (recall the
notations at the beginning of this section)

$\left\\{\begin{array}[c]{ll}dy_{t}=-\big{(}a_{t}y_{t}+\delta_{t}+c^{*}_{t}m^{c}_{t}z^{c}_{t}\big{)}\,dQ_{t}+z_{t}^{\ast}\,dM_{t}+dD_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=\eta.\end{array}\right.$ (6.2)

Since the generator $g^{1}$ is assumed to be uniformly $m$-Lipschitz
continuous with a constant $\widehat{L}_{1}$, it is clear that the mapping
$\widehat{g}^{1}$ is uniformly Lipschitz continuous with the same constant.
Consequently, the $\mathbb{F}$-progressively measurable process $a$ and the
$\mathbb{F}$-predictable process $c$ appearing in (6.2) are bounded,
specifically, $|a_{t}|\leq\widehat{L}_{1}$ and
$\|c_{t}\|\leq\widehat{L}_{1}\sqrt{k}$ for all $t\in[0,T]$. Therefore, the
generator in (6.2) is uniformly $m$-Lipschitz continuous and thus we may apply
Theorem 4.1. Since $b_{t}=\delta_{t}-\zeta^{\ast}_{t}m^{d}_{t}z^{d}_{t}$, we
also have that

$\left\\{\begin{array}[c]{ll}dy_{t}=-\big{(}a_{t}y_{t}+b_{t}+c^{*}_{t}m^{c}_{t}z^{c}_{t}+\zeta^{\ast}_{t}m^{d}_{t}z^{d}_{t}\big{)}\,dQ_{t}+z_{t}^{\ast}\,dM_{t}+dD_{t},\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ y_{T}=\eta.\end{array}\right.$ (6.3)

Since the generators $g^{1}$ and $g^{2}$ satisfy Assumption 2.3 and $\zeta$ is
bounded, we have $b$ belongs to ${\mathcal{H}}^{2}(Q)$.

In the remaining part of the proof, we focus on the linear BSDE (6.3). We
deduce from Proposition 5.1 that, under the present assumptions, if
$(y,z)\in{\mathcal{S}}^{2}$ is a unique solution to (6.3), then the process
$y$ satisfies

$y_{t}=q_{t}^{-1}\,{\mathbb{E}}\bigg{[}q_{T}\eta+\int_{t}^{T}q_{u}b_{u}\,dQ_{u}-\big{(}\Psi^{q,D}_{T}-\Psi^{q,D}_{t}\big{)}\,\Big{|}\,\mathcal{F}_{t}\bigg{]}$
(6.4)

where we denote

$\displaystyle\Psi^{q,D}_{t}:=\int_{0}^{t}q_{u-}\,dD_{u}+\langle
q,D\rangle_{t}$

and the strictly positive process $q$ is given by (5.2). Let us check that,
under the present assumptions, the process $\Psi^{q,D}$ is an
$\mathbb{F}$-supermartingale (which yields that ${\mathbb{E}}\Psi^{q,D}$ is
nonincreasing). Since, by assumption, the process $D$ is nonincreasing, which
implies that $D$ is of finite variation, we have $[q,D]_{t}=\sum_{u\leq
t}\Delta q_{u}\Delta D_{u}$ where $\Delta q_{u}=q_{u-}\Delta\widehat{M}_{u}$.
Since $\Delta\widehat{M}_{u}>-1$ and the process $q$ is strictly positive, it
is easy to see that the process $\Phi$, which is given by

$\displaystyle\Phi_{t}:=\int_{0}^{t}q_{u-}\,dD_{u}+[q,D]_{t}=\int_{0}^{t}q_{u-}\,dD_{u}+\sum_{u\leq
t}\Delta q_{u}\Delta D_{u}$

is nonincreasing since the (pathwise) continuous component of $\Phi$ is
nonincreasing and its jumps satisfy

$\displaystyle\Delta\Phi_{u}=q_{u-}\Delta D_{u}+\Delta q_{u}\Delta
D_{u}=q_{u-}\big{(}1+\Delta\widehat{M}_{u}\big{)}\Delta D_{u}\leq 0.$

Since $\Phi$ is nonincreasing, its dual $\mathbb{F}$-predictable projection
$\Phi^{p}$ is nonincreasing as well. Moreover, it equals

$\displaystyle\Phi^{p}_{t}=\int_{0}^{t}q_{u-}\,dD^{p}_{u}+\langle
q,D\rangle_{t}$

where $D^{p}$ is the dual $\mathbb{F}$-predictable projection of $D$. Note
that since $D^{1}_{T}$ and $D^{2}_{T}$ belong to $L_{2}({\mathcal{F}}_{T})$,
the random variable $D_{T}$ is integrable and thus by noticing that
$\mathcal{F}_{0}$ is trivial, the nonincreasing process $D$ is integrable as
well and thus $D^{p}$ is well defined and the process $M^{D}=D-D^{p}$ is a
uniformly integrable $\mathbb{F}$-martingale (see, for instance, Corollary
5.31 in [14]). Moreover, from Theorem 8.2.19 in [5] and the property that
$D_{T}$ belongs to $L_{2}({\mathcal{F}}_{T})$, we deduce that the random
variable $D^{p}_{T}$ is square-integrable and thus, since
$M^{D}_{t}={\mathbb{E}}(D_{T}-D^{p}_{T}\,|\,\mathcal{F}_{t})$, we see that
$M^{D}$ is a square-integrable $\mathbb{F}$-martingale with $M^{D}_{0}=0$.
Hence, by Assumption 2.2, there exists a process $X\in{\mathcal{L}}^{2}(M)$
such that $M^{D}_{t}=\int_{0}^{t}X^{*}_{u}\,dM_{u}$. We define the process

$\displaystyle
M^{q,D}_{t}:=\int_{0}^{t}q_{u-}\,dM^{D}_{u}=\int_{0}^{t}q_{u-}X^{*}_{u}\,dM_{u}.$

Since the process $q$ belongs ${\mathcal{S}}^{2}$ and $X$ is in
${\mathcal{L}}^{2}(M)$, we obtain

$\displaystyle\mathbb{P}\bigg{[}\int_{0}^{T}\|q_{t-}m_{t}X^{*}_{t}\|^{2}\,dQ_{t}<\infty\bigg{]}\geq\mathbb{P}\bigg{[}\sup_{t\in[0,T]}(q^{2}_{t})\int_{0}^{T}\|m_{t}X^{*}_{t}\|^{2}\,dQ_{t}<\infty\bigg{]}=1$

and thus $M^{q,D}$ is well defined and it is a local $\mathbb{F}$-martingale.
Furthermore, since the process $M^{q,D}$ is bounded from below by the
integrable random variable
$\psi:=-\sup_{t\in[0,T]}(q_{t})(|D_{T}-D_{T}^{p}|+|D_{0}-D_{0}^{p}|)$, it is
in fact an $\mathbb{F}$-supermartingale. To complete the proof of assertion
(a), we observe that, for every $t\in[0,T]$,

$\displaystyle\Psi^{q,D}_{t}=\Phi^{p}_{t}+\int_{0}^{t}q_{u-}\,d(D_{u}-D^{p}_{u})=\Phi^{p}_{t}+\int_{0}^{t}q_{u-}\,dM^{D}_{u}=\Phi^{p}_{t}+M^{q,D}_{t},$

which means that the process $\Psi^{q,D}$ is also an
$\mathbb{F}$-supermartingale. Since we have also assumed that
$\eta=\eta^{1}-\eta^{2}\geq 0$ and the process $b\geq 0,\,\mu_{Q}$-a.e., it is
readily seen from (6.4) that the inequality $y_{t}\geq 0$ holds for all
$t\in[0,T]$. Assertion (b) is also an easy consequence of representation
(6.4). $\Box$

######  Remark 6.1.

It is easy to see that condition (6.1) holds if $g^{1}(\cdot,Y^{2},Z^{2})\geq
g^{2}(\cdot,Y^{2},Z^{2}),\,\mu_{Q}$-a.e., and

$\mu_{t}-\zeta_{t}^{\ast}m^{d}_{t}z^{d}_{t}\geq
0,\quad\mu_{Q}\mbox{\rm-a.e.},$ (6.5)

where we denote

$\displaystyle\mu_{t}:=g^{1}\big{(}t,Y^{2}_{t},\ldots,Z^{2}_{t,k},Z^{1}_{t,k+1},Z^{1}_{t,k+2},\ldots,Z^{1}_{t,d}\big{)}-g^{1}\big{(}t,Y^{2}_{t},\ldots,Z^{2}_{t,k},Z^{2}_{t,k+1},Z^{2}_{t,k+2},\ldots,Z^{2}_{t,d}\big{)}.$

In particular, if the martingale $M$ is continuous so that $k=n$, then
assumption (iv) in Theorem 6.1 reduces to the inequality
$g^{1}(\cdot,Y^{2},Z^{2})\geq g^{2}(\cdot,Y^{2},Z^{2}),\,\mu_{Q}$-a.e., which
is the standard condition in comparison theorems for BSDEs driven by the
Brownian motion (see also Theorem 3.3 in [24] for the case of BSDEs driven by
continuous martingales).

######  Remark 6.2.

Let us assume that $M=(M^{c},M^{d})$ where $M^{c}$ is an
$\mathbb{R}^{n-1}$-valued, continuous $\mathbb{F}$-local martingale and
$M^{d}$ is the compensated martingale of the unit jump at an
$\mathbb{F}$-stopping time $\tau$ with the $\mathbb{F}$-intensity process
$\lambda$. This means that we have, for every $t\in[0,T]$,

$M^{d}_{t}=\mathds{1}_{\\{\tau\leq
t\\}}-\int_{0}^{t}(m^{d}_{u})^{2}\,du=\mathds{1}_{\\{\tau\leq
t\\}}-\int_{0}^{t}\lambda_{u}\,du$

so that $m^{d}_{t}=\sqrt{\lambda_{t}}$. Let us denote
$\zeta_{t}=\psi_{t}m^{d}_{t}=\psi_{t}\sqrt{\lambda_{t}}$. Then condition
$\Delta\widehat{M}_{t}>-1$ is satisfied if $\psi_{t}>-1$ and inequality (6.5)
becomes

$\displaystyle
g^{1}(t,Y^{2}_{t},Z^{2}_{t,1},Z^{1}_{t,2})-g^{1}(t,Y^{2}_{t},Z^{2}_{t,1},Z^{2}_{t,2})\geq\psi_{t}\lambda_{t}(Z^{1}_{t,2}-Z^{2}_{t,2}).$

In particular, we recover condition (25) in Dumitrescu et al. [7] and
condition (12) in [8]. Of course, this condition should be combined with the
postulate that $g^{1}(\cdot,Y^{2},Z^{2})\geq
g^{2}(\cdot,Y^{2},Z^{2}),\,\mu_{Q}$-a.e..

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