Modern Stochastics: Theory and Applications

This paper is concerned with the study of multidimensional generalized backward stochastic differential equations (GBSDEs) with jumps in a general filtration. For convenience of the discussion, let us first make precise the notion of such equations, which is adopted from [9].

Let $T>0$ be a fixed time horizon and consider a filtered probability space $(\varOmega ,\mathcal{F},(\mathcal{F}_{t})_{t\le T},\mathbb{P})$ carrying a standard d-dimensional Brownian motion W and an independent compensated Poisson random measure $\widehat{\pi }$. The filtration $(\mathcal{F}_{t})_{0\le t\le T}$ is assumed to be complete and right continuous. Assume that we are given an ${\mathbb{R}}^{k}$-valued $\mathcal{F}_{T}$-measurable random variable ξ, a random function $f:\varOmega \times [0,T]\times {\mathbb{R}}^{k}\times {\mathbb{R}}^{k\times d}\times {\mathcal{L}_{\lambda }^{2}}\longrightarrow {\mathbb{R}}^{k}$ (see Section 2 for the definition of ${\mathcal{L}_{\lambda }^{2}}$) such that $f(\cdot ,y,z,v)$ is $(\mathcal{F}_{t})$-progressively measurable for each $(y,z,v)$, and an $(\mathcal{F}_{t})_{t\ge 0}$-progressively measurable càdlàg finite-variation process $(R_{t})_{t\in [0,T]}$ such that $R_{0}=0$. Roughly speaking, solving a GBSDE with jumps in a general filtration with terminal time T associated with terminal condition ξ and generator $f+dR$ amounts to finding the usual triple $(Y_{t},Z_{t},V_{t})$ (with Y adapted and Z and V predictable) and a càdlàg martingale $M=(M_{t})_{t\in [0,T]}$ that is orthogonal to W and $\widehat{\pi }$ (see Lemma 1) such that the following equation is satisfied $\mathbb{P}$-a.s.: This equation is usually denoted by GBSDE$(\xi ,f+dR)$. Note that the reason behind adding the martingale M to the definition of GBSDE (1) is the fact that we do not assume that the underlying filtration is generated by W and $\widehat{\pi }$, and in such cases, the martingale representation property may fail.

Nonlinear BSDEs with jumps (i.e., the underlying filtration is generated by a Brownian motion and an independent Poisson random measure) were first introduced by Tang and Li [19]. They proved the existence and uniqueness of the solution under a Lipschitz continuity condition on the generator w.r.t. the variables. Since then, a lot of papers (see, e.g., [13, 1, 17, 20, 16, 12, 9], and the references therein) and books (see, e.g., [18] and [5]) studied BSDEs with jumps due to the connections of this subject with mathematical finance (see, e.g., [5]) (e.g., if the Brownian motion stands for the noise from a financial market, then the Poisson random measure can be interpreted as the randomness of the insurance claims), stochastic control (see, e.g., [10]), and partial differential equations (see, e.g., [1]), and so on. Since the work of Tang and Li [19], the attempts of generalization of their results have been made in several different directions. First of all, many papers aimed at relaxing the Lipschitz condition on the generator w.r.t. y. For example, Pardoux [13] considered a monotonicity condition on the generator w.r.t. y and a linear growth condition on y. Some efforts were devoted to weaken the square integrability on the coefficients, for example, $\mathbb{E}[|\xi {|}^{2}+{\int _{0}^{T}}|f(t,0,0,0){|}^{2}dt]<+\infty $. Yao [20] gave the existence and uniqueness results for ${\mathbb{L}}^{p}$-solutions ($p>1$) for BSDEs with jumps for a monotonic generator (not the same monotonicity condition considered in our paper) and ${\mathbb{L}}^{p}$ coefficients. Later, Li and Wei [12] analyzed fully coupled BSDEs with jumps and showed the existence and uniqueness of ${\mathbb{L}}^{p}$-solutions ($p\ge 2$) for such equations for a monotone generator and p-integrable data. Further, other settings of BSDEs with jumps have been introduced. Pardoux [13] studied a class of BSDEs with jumps called generalized BSDEs with jumps that involves an integral w.r.t. an increasing continuous process. The author shows the uniqueness and existence of the GBSDE with generator monotone in y and square-integrable data.

Recently, Kruse and Popier [9] considered another direction of generalization concerning the underlying filtration, which is no longer assumed to be generated by W and $\widehat{\pi }$. In fact, they studied multidimensional BSDEs in a general filtration of type (1) with $R\equiv 0$. The authors established the existence and uniqueness of ${\mathbb{L}}^{p}$-solutions ($p>1$) under a monotonicity assumption on the generator f w.r.t. y and under the condition that the data ξ and $f(t,0,0,0)$ are in ${\mathbb{L}}^{p}$, $p>1$, that is, Moreover, they also considered the case of a random terminal time that is not necessarily bounded.

In our paper, we first investigate the existence and uniqueness of ${\mathbb{L}}^{p}(p\ge 2)$-solutions (see Definition 1) for GBSDEs (1) in a deterministic time horizon. We suppose that f is monotonic w.r.t. y (this condition is essential in the study of BSDEs with random terminal time), Lipschitz continuous w.r.t. to z and v, and satisfies a very general growth condition w.r.t. y considered earlier in the Brownian setting in [4] and recently in the case of jumps in [9]: This condition seems to be the best possible growth condition on f w.r.t. y and is widely used in the theory of partial differential equations (see [2] and the references therein).

Concerning the data, we assume that a p-integrability condition is satisfied (see assumption $(H1)$). Moreover, under an additional assumption on the jump component of f (see $(H{6^{\prime }})$ in Section 4), we provide a comparison principle in dimension one (see the counterexample in [1]). Then, we extend the results obtained in the case of deterministic time horizon to the case of a random terminal time that is not necessarily bounded.

Let us highlight the main contribution of the paper compared to the existing literature. On the one hand, our results extend the work of Kruse and Popier [9] to the case of generalized BSDEs. Furthermore, we strengthen their results even in the case $R=0$ since our p-integrability condition on $f(t,0,0,0)$ (see assumption $(H1)$) is weaker than the ${\mathbb{L}}^{p}$-integrability (2) assumed in their paper. It should be mentioned that, due to the p-integrability assumed on $f(t,0,0,0)$ and also to the process $dR$, some difficulties arise. Indeed, as in [4] and [9], to study the ${\mathbb{L}}^{p}$-solutions, a result on the existence and uniqueness in the classical ${\mathbb{L}}^{2}$ case (see Theorem 1) is first needed. To obtain such a result, the main trick is to truncate the coefficients with suitable truncation functions in order to have a bounded solution Y, which is a key tool in the proof in the ${\mathbb{L}}^{2}$ case (see [14, Prop. 2.4 and Thm. 2.2], Proposition 2.2 in [3] used in [4, Thm. 4.2] and [9, Lemma 4 and Thm. 1]). The approach followed in the papers mentioned to get this important estimate fails in our context. This is the reason why we give nonstandard estimates in Lemma 2, which allow us to overcome this problem.

On the other hand, we generalize the work of Pardoux [13] to the situation of a general filtration. Moreover, even in the case of a Wiener–Poisson filtration (filtration generated by W and π), compared to [13], we weaken the growth condition on f w.r.t. y stated in assumption (3), instead of the linear growth condition on the variable y, and derive the existence and uniqueness of ${\mathbb{L}}^{p}$-solutions of our GBSDE, whereas only the classical ${\mathbb{L}}^{2}$-solutions were studied in [13]. Note also that, in our case, GBSDE involves an integral w.r.t. a finite-variation càdlàg process unlike [13], where an integral w.r.t. a continuous increasing process is considered instead.

Our main motivation for writing this paper is because it is a first step in the study of our future work on the existence and uniqueness of ${\mathbb{L}}^{p}$-solutions for reflected GBSDEs. Note that since we solve that problem using a penalization method, the comparison principle obtained here is primordial. Finally, to the best of our knowledge, there is no such result in the literature.

The rest of the paper is organized as follows: in the following section, we give the mathematical setting of this paper and some basic identities. In Section 3, we study the existence and uniqueness of ${\mathbb{L}}^{p}$-solutions on a fixed time interval, which is done in three parts. First, we study the classical case of ${\mathbb{L}}^{2}$-solutions. The proof method follows the arguments and techniques (convolution, weak convergence, truncation technique) given in [8, 13, 14, 4, 9] with obvious modifications. Then, in the remaining parts, we extend the result to ${\mathbb{L}}^{p}$-solutions for any $p\ge 2$, using the right a priori estimate, the ${\mathbb{L}}^{2}$ case result, and a truncation technique. In Section 4, a comparison principle for GBSDEs with jumps in dimension 1 is provided. Finally, Section 5 is devoted to the case of a random terminal time.

Throughout this paper, $T>0$ is a fixed time horizon, $(\varOmega ,\mathcal{F},(\mathcal{F}_{t})_{t\le T},\mathbb{P})$ is a filtered probability space. The filtration $(\mathcal{F}_{t},0\le t\le T)$ is assumed to be complete and right continuous. We suppose that $(\varOmega ,\mathcal{F},(\mathcal{F}_{t})_{t\le T},\mathbb{P})$ supports a d-dimensional Wiener process $(W_{t},0\le t\le T)$ and a random Poisson measure π on ${\mathbb{R}}^{+}\times U$, where $U:={\mathbb{R}}^{n}\setminus \{0\}$ is equipped with its Borel field $\mathcal{U}$, with the compensator $\nu (dt,de)=dt\lambda (de)$ such that $\{\widehat{\pi }([0,t]\times A)=(\pi -\nu )([0,t]\times A)\}_{t\le T}$ is a martingale for all $A\in \mathcal{U}$ satisfying $\lambda (A)<+\infty $. Here, λ is assumed to be a σ-finite Lévy measure on $(U,\mathcal{U})$ such that

Let $\mathcal{P}$ denote the σ-algebra of predictable sets on $\varOmega \times [0,T]$, and let us introduce the following notation: For a given process $R\in \mathcal{V}$, we denote by $|R|_{t}$ the variation of R on $[0,t]$ and by $dR$ the random measure generated by its trajectories. By $\mathcal{T}$ we denote the set of all stopping times with values in $[0,T]$ and by $\mathcal{T}_{t}$ the set of all stopping times with values in $[t,T]$. We say that a sequence $(\tau _{k})_{k\in \mathbb{N}}\subset T$ is stationary if $\mathbb{P}(\liminf _{k\to +\infty }\{\tau _{k}=T\})=1$. For $X\in \mathcal{S}$, we set $X_{t-}=\lim _{s\nearrow t}X_{s}$ and $\Delta X_{t}=X_{t}-X_{t-}$ with the convention that $X_{0-}=0$.

Now, since we are dealing with a general filtration, we recall Lemma III.4.24 in [7], which gives the representation property of a local martingale in our context.

The Euclidean norm of a vector $y\in {\mathbb{R}}^{k}$ will be defined by $|y|=\sqrt{{\sum _{i=1}^{k}}|y_{i}{|}^{2}}$, and for any $k\times d$ matrix z, we define $|z|=\sqrt{\operatorname{Trace}(z{z}^{t})}$, where ${z}^{t}$ stands for the transpose of z. The quadratic variation of a martingale $M\in {\mathbb{R}}^{k}$ is defined by $[M]_{t}={\sum _{i=1}^{k}}[{M}^{i}]_{t}$. By ${[M]}^{c}$ we denote the continuous part of the quadratic variation $[M]$. Let us introduce the following spaces of processes for any real constant $p\ge 2$: In what follows, let ξ be an ${\mathbb{R}}^{k}$-valued and $\mathcal{F}_{T}$-measurable random variable, and let R be a process in $\mathcal{V}$. Finally, let us consider a random function $f:[0,T]\times \varOmega \times {\mathbb{R}}^{k}\times {\mathbb{R}}^{k\times d}\times {\mathcal{L}_{\lambda }^{2}}\longrightarrow {\mathbb{R}}^{k}$ measurable with respect to $\mathcal{P}\otimes \mathcal{B}({\mathbb{R}}^{k})\otimes \mathcal{B}({\mathbb{R}}^{k\times d})\otimes \mathcal{B}({\mathcal{L}_{\lambda }^{2}})$. In the paper, we consider the following hypotheses:

To begin with, let us make precise the notion of ${\mathbb{L}}^{p}$-solutions of the GBSDE (1), which we consider throughout this paper.

In this section, we assume that $k=1$ and aim at showing a comparison theorem for GBSDE. Our result, in particular, extends to the case of generalized BSDEs in a general filtration the comparison theorem given in [9, Prop. 4]. We follow the argument of [16]. In particular, we consider the Doléans–Dade exponential local martingale. Let α, β be predictable processes integrable w.r.t. $dt$ and $dW_{t}$, respectively. Let γ be a predictable process defined on $[0,T]\times \varOmega \times \mathbb{R}$ integrable w.r.t. $\widehat{\pi }(de,ds)$. For any $0\le t\le s\le T$, let E be the solution of and let Γ be the solution of Of course, $\varGamma _{t,s}=\exp ({\int _{t}^{s}}\alpha _{r}dr)E_{t,s}$, and with $X_{t}={\int _{0}^{t}}\int _{U}u\pi (du,ds)$.

Note that, classically, if $\gamma _{t}(e)\ge -1,\hspace{2.5pt}d\mathbb{P}\otimes ds\otimes d\lambda (e)$-a.s., then $\varGamma _{t,.}\ge 0$ a.s. (see [16, Prop. 3.1]).

We make the following monotonicity assumption on f w.r.t. v

Notice that $(H{6^{\prime }})$ implies $(H6)$ (see Section 5 in [9]).

We begin by showing that a linear GBSDE with jumps can be written as a conditional expectation via an exponential semimartingale. This result will be used to prove the comparison theorem.

Notice that assumptions $(H1)$–$(H6)$ made on $f_{1}$ are imposed only to ensure the existence of a solution $({Y}^{1},{Z}^{1},{V}^{1},{M}^{1})$. The following corollary, which follows immediately from Proposition 1, gives again a uniqueness result for GBSDE (1) in ${\varXi }^{p}$ in dimension 1.

In this section, we study the issue of existence and uniqueness of ${\mathbb{L}}^{p}(p\ge 2)$-solutions of GBSDEs with random terminal time. We follow the approach in [14, Section 4]. Let τ be an $\mathcal{F}$-stopping time, not necessarily bounded. Assumptions considered in the case of GBSDE with constant time (precisely, $(H2)$, $(H3)$, $(H5)$, and $(H6)$) still hold except for $(H4)$ and $(H1)$, for which we give the analogues for $p\ge 2$:

Finally, we will need the following additional assumption on ξ and f:

$(H7)$ ξ is $\mathcal{F}_{\tau }$-measurable, and $\mathbb{E}[{({\int _{0}^{\tau }}{e}^{\rho \tau }|f(t,\xi _{t},\eta _{t},\gamma _{t})|ds)}^{p}]<+\infty $, where $\xi _{t}=\mathbb{E}(\xi |\mathcal{F}_{t})$ and $(\eta ,\gamma ,N)$ are given by the martingale representation with N orthogonal to W and $\widetilde{\pi }$. Moreover, the following holds:

Next, let us make precise the notion of a solution of GBSDE with random terminal time.