VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA73 10.15559/17-VMSTA73 Research Article Lp(p≥2) -solutions of generalized BSDEs with jumps and monotone generator in a general filtration EddahbiM’hamedm.eddahbi@uca.maa FakhouriImadeimadefakhouri@gmail.coma∗ OuknineYoussefouknine@uca.ac.maab Cadi Ayyad University, Av. Abdelkrim Khattabi, 40000, Guéliz–Marrakesh, Morocco Hassan II Academy of Sciences and Technology, Morocco Corresponding author. 2017 722017412563 2842016 1812017 1912017 © 2017 The Author(s). Published by VTeX2017 Open access article under the CC BY license.

In this paper, we study multidimensional generalized BSDEs that have a monotone generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. First, we prove the existence and uniqueness of Lp(p≥2) -solutions in the case of a fixed terminal time under suitable p-integrability conditions on the data. Then, we extend these results to the case of a random terminal time. Furthermore, we provide a comparison result in dimension 1.

 Generalized backward stochastic differential equations (GBSDEs) with jumps Lp solution monotone generator comparison theorem 60H10 60H20 60F25 Introduction

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 0$]]> be a fixed time horizon and consider a filtered probability space (Ω,F,(Ft)t≤T,P) carrying a standard d-dimensional Brownian motion W and an independent compensated Poisson random measure πˆ . The filtration (Ft)0≤t≤T is assumed to be complete and right continuous. Assume that we are given an Rk -valued FT -measurable random variable ξ, a random function f:Ω×[0,T]×Rk×Rk×d×Lλ2⟶Rk (see Section 2 for the definition of Lλ2 ) such that f(·,y,z,v) is (Ft) -progressively measurable for each (y,z,v) , and an (Ft)t≥0 -progressively measurable càdlàg finite-variation process (Rt)t∈[0,T] such that R0=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 (Yt,Zt,Vt) (with Y adapted and Z and V predictable) and a càdlàg martingale M=(Mt)t∈[0,T] that is orthogonal to W and πˆ (see Lemma 1) such that the following equation is satisfied  P -a.s.: Yt=ξ+∫tTf(s,Ys,Zs,Vs)ds+∫tTdRs−∫tTZsdWs−∫tT∫UVs(e)πˆ(de,ds)−∫tTdMs,t∈[0,T]. This equation is usually denoted by GBSDE (ξ,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  πˆ , 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, E[|ξ|2+∫0T|f(t,0,0,0)|2dt]<+∞ . Yao [20] gave the existence and uniqueness results for Lp -solutions ( p>1 1$]]>) for BSDEs with jumps for a monotonic generator (not the same monotonicity condition considered in our paper) and Lp coefficients. Later, Li and Wei [12] analyzed fully coupled BSDEs with jumps and showed the existence and uniqueness of Lp -solutions ( p≥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 πˆ . In fact, they studied multidimensional BSDEs in a general filtration of type (1) with R≡0 . The authors established the existence and uniqueness of Lp -solutions ( p>1 1$]]>) under a monotonicity assumption on the generator f w.r.t. y and under the condition that the data ξ and f(t,0,0,0) are in Lp , p>1 1$]]>, that is, E[|ξ|p+∫0T|f(t,0,0,0)|pdt]<+∞. 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 Lp(p≥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]: ∀r>0,sup|y|≤r|f(·,y,0,0)−f(·,0,0,0)|∈L1(Ω×[0,T]). 0,\hspace{1em}\underset{|y|\le r}{\sup }\big|f(\cdot ,y,0,0)-f(\cdot ,0,0,0)\big|\in {\mathbb{L}}^{1}\big(\varOmega \times [0,T]\big).\]]]> 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 (H6′) 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 Lp -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 Lp -solutions, a result on the existence and uniqueness in the classical L2 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 L2 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 Lp -solutions of our GBSDE, whereas only the classical L2 -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 Lp -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 Lp -solutions on a fixed time interval, which is done in three parts. First, we study the classical case of L2 -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 Lp -solutions for any p≥2 , using the right a priori estimate, the L2 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.

 Preliminaries

Throughout this paper, T>0 0$]]> is a fixed time horizon, (Ω,F,(Ft)t≤T,P) is a filtered probability space. The filtration (Ft,0≤t≤T) is assumed to be complete and right continuous. We suppose that (Ω,F,(Ft)t≤T,P) supports a d-dimensional Wiener process (Wt,0≤t≤T) and a random Poisson measure π on R+×U , where U:=Rn∖{0} is equipped with its Borel field U , with the compensator ν(dt,de)=dtλ(de) such that {πˆ([0,t]×A)=(π−ν)([0,t]×A)}t≤T is a martingale for all A∈U satisfying λ(A)<+∞ . Here, λ is assumed to be a σ-finite Lévy measure on (U,U) such that ∫U(1∧|e|2)λ(de)<∞.

Let P denote the σ-algebra of predictable sets on Ω×[0,T] , and let us introduce the following notation:

S is the set of all adapted càdlàg processes.

Gloc(π) is the set of P×U -measurable functions V on Ω×[0,T]×U such that, for any t≥0 , ∫0t∫U(|Vs(e)|2∧|Vs(e)|)λ(de)(ds)<+∞a.s.

H (resp. H(0,T) ) is the set of all predictable processes on R+ (resp. on [0,T] ). Lloc2(W) is the subspace of H such that, for any t≥0 , ∫0t|Zs|2ds<+∞a.s.

Mloc is the set of càdlàg local martingales orthogonal to W and πˆ . If M∈Mloc , then [M,Wi]t=0,1≤i≤d,and[M,πˆ(A,·)]t=0, for all A∈U . In other words, E(ΔM∗π∣P⊗U)=0 , where the product ∗ denotes the integral process (see II.1.5 in [7]).

M is the subspace of Mloc of martingales.

V is the set of all càdlàg progressively measurable processes R of finite variation such that R0=0 .

 For a given process R∈V , we denote by |R|t the variation of R on [0,t] and by dR the random measure generated by its trajectories. By T we denote the set of all stopping times with values in [0,T] and by Tt the set of all stopping times with values in [t,T] . We say that a sequence (τk)k∈N⊂T is stationary if P(lim infk→+∞{τk=T})=1 . For X∈S , we set Xt−=lims↗tXs and ΔXt=Xt−Xt− with the convention that X0−=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.

Every local martingale has a decomposition ∫0.ZsdWs+∫0.∫UVs(e)πˆ(de,ds)+M, where M∈Mloc , Z∈Lloc2(W) , and V∈Gloc(π) .

The Euclidean norm of a vector y∈Rk will be defined by |y|=∑i=1k|yi|2 , and for any k×d matrix z, we define |z|=Trace(zzt) , where zt stands for the transpose of z. The quadratic variation of a martingale M∈Rk is defined by [M]t=∑i=1k[Mi]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≥2 :

Lp is the space of Rk -valued random variables ξ such that ‖ξ‖Lp:=E[|ξ|p]1/p<+∞.

Sp is the space of Rk -valued, Ft -adapted, and càdlàg processes (Yt)0≤t≤T such that ‖Y‖Sp:=E[sup0≤t≤T|Yt|p]1/p<+∞.

Mp is the set of all Rk -valued martingales M∈M such that E([M]T)p2<+∞ .

Mp is the set of Rk×d -valued and F -progressively measurable processes (Zt)0≤t≤T such that ‖Z‖Mp:=E[(∫0T|Zs|2ds)p2]1/p<+∞.

Lp is the set of P⊗U -measurable mappings V:Ω×[0,T]×U→Rk such that ‖V(e)‖Lp:=E[(∫0T∫U|Vs(e)|2λ(de)ds)p2]1/p<+∞.

Lλp is the set of measurable functions ϕ:U→Rk such that ‖ϕ(e)‖Lλp:=(∫U|ϕ(e)|pλ(de))1/p<+∞.

Ξp is the space Sp×Mp×Lp×Mp .

Vp is the set of all processes R∈V such that ‖R‖Vp:=E(|R|Tp)1/p<∞ , where |R|T denotes the total variation of R on [0,T] .

 In what follows, let ξ be an Rk -valued and FT -measurable random variable, and let R be a process in V . Finally, let us consider a random function f:[0,T]×Ω×Rk×Rk×d×Lλ2⟶Rk measurable with respect to P⊗B(Rk)⊗B(Rk×d)⊗B(Lλ2) . In the paper, we consider the following hypotheses:

E[|ξ|p+(∫0T|f(t,0,0,0)|dt)p+|R|Tp]<+∞ .

For every (t,z,v)∈[0,T]×Rk×d×Lλ2 , the mapping y∈Rk→f(t,y,z,v) is continuous.

There exists μ∈R such that (f(t,y,z,v)−f(t,y′,z,v))(y−y′)≤μ(y−y′)2, for all t∈[0,T],y,y′∈Rk,z∈Rk×d,v∈Lλ2 .

For every r>0 0$]]>, the mapping t∈[0,T]→sup|y|≤r|f(t,y,0,0)−f(t,0,0,0)| belongs to L1(Ω×[0,T]) .

f is Lipschitz continuous w.r.t. z, that is, there exists a constant L>0 0$]]> such that |f(t,y,z,v)−f(t,y,z′,v)|≤L|z−z′|, for all t∈[0,T],y∈Rk,z,z′∈Rk×d,v∈Lλ2 .

f is Lipschitz continuous w.r.t. v, that is, there exists a constant L>0 0$]]> such that |f(t,y,z,v)−f(t,y,z,v′)|≤L‖v−v′‖Lλ2, for all t∈[0,T],y∈Rk,z∈Rk×d,v,v′∈Lλ2 .

To begin with, let us make precise the notion of Lp -solutions of the GBSDE (1), which we consider throughout this paper.

We say that (Y,Z,V,M):=(Yt,Zt,Vt,Mt)0≤t≤T is an Lp -solution of the GBSDE (1) if (Y,Z,V,M)∈Ξp and (1) is satisfied P -a.s.

 Generalized BSDEs with constant terminal time <inline-formula id="j_vmsta73_ineq_174"><alternatives> <mml:math><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math> <tex-math><![CDATA[${\mathbb{L}}^{2}$]]></tex-math></alternatives></inline-formula>-solutions

In this subsection, we study the classical case of L2 -solutions of GBSDE (1). The results given here generalize those of [13] and [9]. Note that the integrability condition (H1)p=2 made on f(·,0,0,0) is weaker than the assumption E∫0T|f(t,0,0,0)|2dt<+∞ , t∈[0,T] , made in those papers, which means that our assumption is weaker than that of [9] even in the case R≡0 .

Let us begin by giving nonstandard a priori estimates on the solution, which will play a primordial role in the proof of Theorem 1. Let us first make the following assumption:

There exist L≥0 , μ∈R , and a nonnegative progressively measurable process {ft}t∈[0,T] satisfying E(∫0Tfsds)2<+∞ such that

∀(t,y,z,v)∈[0,T]×Rk×Rk×d×Lλ2 , sgnˆ(y)f(t,y,z,v)≤ft+μ|y|+L|z|+L‖v‖Lλ2,dt⊗dP-a.s.

Note that (A) is not a new assumption, but a direct consequence of assumptions (H3) , (H5) , and (H6) with ft=|f(t,0,0,0)| . In fact, three assumptions (H3) , (H5) , and (H6) are reduced to a single one (assumption  (A) ) for simplicity.

Let assumption (A) hold, and let (Y,Z,V,M) be a solution of GBSDE (1). If Y∈S2 and E|ξ|2+E(∫0Tfsds)2+E|R|T2<+∞, then, (Z,V,M) belongs to M2×L2×M2 , and for some a≥μ+2L2 , there is a constant C>0 0$]]> such that, for all 0≤q≤t≤T , E[sups∈[t,T]e2as|Ys|2+∫tTe2as|Zs|2ds+∫tT∫Ue2as|Vs(e)|2λ(de)ds+e2aT[M]T−e2at[M]t|Fq]≤CE[e2aT|ξ|2+(∫tTeasfsds)2+(∫tTeasd|R|s)2|Fq].

The proof is performed in two steps. For simplicity, we can assume w.l.o.g. that a=0 . Indeed, let us fix a≥μ+2L2 and define Y˜t=eatYt , Z˜=eatZt , V˜t=eatVt , dM˜t=eatdMt . Observe that (Y˜,Z˜,V˜,M˜) solves the following GBSDE: Y˜t=ξ˜+∫tTf˜(s,Y˜s,Z˜s,V˜s)ds+∫tTdR˜s−∫tTZ˜sdWs−∫tT∫UV˜s(e)πˆ(de,ds)−∫tTdM˜s,t∈[0,T], where ξ˜=eaTξ , f˜(t,y,z,v)=eatf(t,e−aty,e−atz,e−atv)−ay , and dR˜t=eatdRt . Notice that f˜ satisfies assumption (A) with f˜t=eatft , μ˜=μ−a , L˜=L . Since we are working on a compact time interval, the integrability conditions are equivalent with or without the superscript ~. Thus, with this change of variable, we reduce to the case a=0 and μ+2L2≤0 . We omit the superscript ~ for notational convenience.

Step 1. First, we show that there exists a constant C>0 0$]]> such that, for all 0≤q≤t≤T , E(∫tT|Zs|2ds+∫tT∫U|Vs(e)|2λ(de)ds+∫tTd[M]s|Fq)≤CE(supu∈[t,T]|Yu|2+(∫tTfsds)2+(∫tTd|R|s)2|Fq). Since there is a lack of integrability of the processes (Z,V,M) , we are proceeding by localization. For n∈N , we set τn=inf{t>0;∫0t|Zs|2ds+∫0t∫U|Vs(e)|2λ(de)ds+[M]t>n}∧T. 0;{\int _{0}^{t}}|Z_{s}{|}^{2}ds+{\int _{0}^{t}}\int _{U}{\big|V_{s}(e)\big|}^{2}\lambda (de)ds+[M]_{t}>n\Bigg\}\wedge T.\]]]> By Itô’s formula (see [15, Thm. II.32]), |Yt∧τn|2+∫t∧τnτn|Zs|2ds+∫t∧τnτn∫U|Vs(e)|2π(de,ds)+∫t∧τnτnd[M]s=|Yτn|2+2∫t∧τnτnYsf(s,Ys,Zs,Vs)ds+2∫t∧τnτnYs−dRs−2∫t∧τnτnYsZsdWs−2∫t∧τnτn∫UYs−Vs(e)πˆ(de,ds)−2∫t∧τnτnYs−dMs. But from (A) , the basic inequality 2ab≤2a2+b22 , and the fact that μ+2L2≤0 we have that 2Ysf(s,Ys,Zs,Vs)≤2L|Ys||Zs|+2L|Ys|‖Vs(e)‖Lλ2+2μ|Ys|2+2|Ys|fs≤2(μ+2L2)|Ys|2+2|Ys|fs+12|Zs|2+12∫U|Vs(e)|2λ(de)≤2|Ys|fs+12|Zs|2+12∫U|Vs(e)|2λ(de). Then, plugging the last inequality into (8), we deduce 12∫t∧τnτn|Zs|2ds+∫t∧τnτn∫U|Vs(e)|2π(de,ds)−12∫t∧τnτn∫U|Vs(e)|2λ(de)ds+∫t∧τnτnd[M]s≤sups∈[t∧τn,T]|Ys|2+2sups∈[t∧τn,T]|Ys|(∫t∧τnTfsds)+2sups∈[t∧τn,T]|Ys|(∫t∧τnTd|R|s)−2∫t∧τnτnYsZsdWs−2∫t∧τnτn∫UYs−Vs(e)πˆ(de,ds)−2∫t∧τnτnYs−dMs. Hence, using the inequality 2ab≤a2+b2 , we obtain 12∫t∧τnτn|Zs|2ds+∫t∧τnτn∫U|Vs(e)|2π(de,ds)−12∫t∧τnτn∫U|Vs(e)|2λ(de)ds+∫t∧τnτnd[M]s≤3supu∈[t∧τn,T]|Yu|2+(∫t∧τnTfsds)2+(∫t∧τnTd|R|s)2−2∫t∧τnτnYsZsdWs−2∫t∧τnτn∫UYs−Vs(e)πˆ(de,ds)−2∫t∧τnτnYs−dMs. Note that since Y∈S2 , by the definition of the stopping time τn it follows by the BDG inequality that ∫0t∧τnYsZsdWs , ∫0t∧τn∫UYs−Vs(e)πˆ(de,ds) and ∫0t∧τnYs−dMs are uniformly integrable martingales. Consequently, taking the conditional expectation w.r.t. Fq , 0≤q≤t≤T , in both sides of (10) yields E(12∫t∧τnτn|Zs|2ds+12∫t∧τnτn∫U|Vs(e)|2λ(de)ds+∫t∧τnτnd[M]s|Fq)≤E(3supu∈[t∧τn,T]|Yu|2+(∫t∧τnTfsds)2+(∫t∧τnTd|R|s)2|Fq). Therefore, letting n to infinity and using Fatou’s lemma, we obtain (7).

Step 2. In this step, we will estimate E(supu∈[t,T]|Yu|2|Fq) , 0≤q≤t≤T . Applying Itô’s formula to |Yt|2 for each t∈[0,T] , we get |Yt|2+∫tT|Zs|2ds+∫tT∫U|Vs(e)|2π(de,ds)+∫tTd[M]s=|ξ|2+2∫tTYsf(s,Ys,Zs,Vs)ds+2∫tTYs−dRs−2∫tTYsZsdWs−2∫tT∫UYs−Vs(e)πˆ(de,ds)−2∫tTYs−dMs, but in view of (9), we deduce |Yt|2+12∫tT|Zs|2ds+∫tT∫U|Vs(e)|2π(de,ds)+∫tTd[M]s≤|ξ|2+2∫tT|Ys|fsds+2∫tT|Ys−|d|R|s+12∫tT∫U|Vs(e)|2λ(de)ds−2∫tTYsZsdWs−2∫tT∫UYs−Vs(e)πˆ(de,ds)−2∫tTYs−dMs. Recalling that Y∈S2 , thanks to the first step (estimate (7)), it follows that Z∈M2 , V∈L2 , and M∈M2 . Therefore, by the BDG inequality we deduce that ∫0tYsZsdWs , ∫0t∫UYs−Vs(e)πˆ(de,ds) , and ∫0tYs−dMs are uniformly integrable martingales. Therefore, taking the conditional expectation in (12) w.r.t. Fq , it follows that, for all 0≤q≤t≤T , E(12∫tT|Zs|2ds+12∫tT∫U|Vs(e)|2λ(de)ds+∫tTd[M]s|Fq)≤E(ς∣Fq), where ς=|ξ|2+2∫tT|Ys|fsds+2∫tT|Ys−|d|R|s .

Next, we deduce from (12) that supu∈[t,T]|Yu|2≤ς+12∫tT∫U|Vs(e)|2λ(de)ds+2supu∈[t,T]|∫uTYsZsdWs|+2supu∈[t,T]|∫uT∫UYs−Vs(e)πˆ(de,ds)|+2supu∈[t,T]|∫uTYs−dMs|. Consequently, taking the conditional expectation w.r.t. Fq , 0≤q≤t≤T , we obtain from (13) that E(supu∈[t,T]|Yu|2∣Fq)≤E(2ς+2supu∈[t,T]|∫uTYsZsdWs|+2supu∈[t,T]|∫uT∫UYs−Vs(e)πˆ(de,ds)|+2supu∈[t,T]|∫uTYs−dMs||Fq). Next, applying the BDG inequality to the martingale terms implies that there exists a constant C1>0 0$]]> such that 2E(supu∈[t,T]|∫uTYsZsdWs||Fq)≤2C1E(supu∈[t,T]|Yu|(∫tT|Zs|2ds)1/2|Fq)≤14E(supu∈[t,T]|Yu|2|Fq)+2C12E(∫tT|Zs|2ds|Fq), 2E(supu∈[t,T]|∫uTYs−Vs(e)πˆ(de,ds)||Fq)≤2C1E(supu∈[t,T]|Yu|(∫tT|Vs(e)|2π(de,ds))1/2|Fq)≤14E(supu∈[t,T]|Yu|2|Fq)+2C12E(∫tT|Vs(e)|2λ(de)ds|Fq), and 2E(supu∈[t,T]|∫uTYs−dMs||Fq)≤2C1E(supu∈[t,T]|Yu|(∫tTd[M]s)1/2|Fq)≤14E(supu∈[t,T]|Yu|2|Fq)+2C12E(∫tTd[M]s|Fq). Hence, plugging estimates (16)–(18) into (15) implies in view of (13) that there exits a constant C2>0 0$]]> such that E(supu∈[t,T]|Yu|2|Fq)≤C2E(ς∣Fq). Applying Young’s inequality yields C2E(∫tT|Ys|fsds+∫tT|Ys−|d|R|s|Fq)≤14E(supu∈[t,T]|Yu|2|Fq)+C3E[(∫tTfsds)2+(∫tTd|R|s)2|Fq], from which we deduce, coming back to the definition of ς, that there exists C4>0 0$]]> such that E(supu∈[t,T]|Yu|2|Fq)≤C4E(|ξ|2+(∫tTfsds)2+(∫tTd|R|s)2|Fq). Finally, combining this with (7), the desired result follows, which ends the proof.  □

Now, we give the main result of this subsection.

(L2 -solutions) Assume that (H1)p=2 – (H6) are in force. Then, there exists a unique L2 -solution (Y,Z,V,M) for the GBSDE (1).

Uniqueness. Let (Y,Z,V,M) and (Y′,Z′,V′,M′) denote respectively two L2 -solutions of GRBSDE (1). Define (Y¯,Z¯,V¯,M¯)=(Y−Y′,Z−Z′,V−V′,M−M′) . Then, (Y¯,Z¯,V¯,M¯) solves the following GBSDE in Ξ2 : Y¯t=∫tT(f(s,Ys,Zs,Vs)−f(s,Ys′,Zs′,Vs′))ds−∫tTZ¯sdWs−∫tT∫UV¯s(e)πˆ(de,ds)−∫tTdM¯s,t∈[0,T]. It follows from (H3) , (H5) , and (H6) that sgnˆ(y¯)(f(t,y,z,v)−f(t,y′,z′,v′))=sgnˆ(y¯)[f(t,y,z,v)−f(t,y′,z,v)+f(t,y′,z,v)−f(t,y′,z′,v)+f(t,y′,z′,v)−f(t,y′,z′,v′)]≤μ|y|+L|z|+L‖v‖Lλ2, which means that assumption (A) is satisfied for the generator of GBSDE (20) with ft≡0 . By Lemma 2 with q=t=0 we obtain immediately that (Y¯,Z¯,V¯,M¯)=(0,0,0,0) . The proof of the uniqueness is then complete.

Existence. Before giving the proof of the existence part, we will talk a little bit about it, but let us first give the following assumption, the so-called general growth condition, which will be needed later: (Hgg)For every(t,y)∈[0,T]×R,|f(t,y,0,0)|≤|f(t,0,0,0)|+γ(|y|), where γ:R+→R+ is a deterministic continuous increasing function.

The proof method of Theorem 1 is enlightened by [8, 13, 14, 4, 9], but of course with some obvious changes. More precisely, the first step uses arguments given in [8, 13, 9], whereas the techniques used in the second step, the convolution and weak convergence, are borrowed from [13]. The truncation techniques applied in the third and fourth steps are taken partly from [4, 9, 8]. However, it should be mentioned that since we have changed, compared to [9], the L2 -integrability condition of f(t,0,0,0) from E∫0T|f(t,0,0,0)|2dt<+∞ to the one given in (H1)p=2 and due also to the finite-variation part dR , some new troubles come up, especially, when we want to prove an analogous result of [9, Lemma 4], which says that whenever the data is bounded, so is the solution of the GBSDE. Their approach fails in our context. This is the reason why we give nonstandard estimates in Lemma 2, which allows us to overcome this problem (see, e.g., estimates (26) and (28)). Additionally, in order to prove the existence part of Theorem 1, we first need an existence result under the assumptions of this theorem but with (H4) replaced with (Hgg) , which extends results given in [13] and [9].

The proof is divided into five steps as follows. Note that we will frequently apply Lemma 2. For simplicity, we will assume w.l.o.g. that a=0 , which means that, in this case, μ<0 (since μ+2L2≤a=0 , i.e., μ≤−2L2<0 ). In the rest of the proof, we will assume that μ<0 .

Step 1. We first assume additionally that there exists a constant l>0 0$]]> such that |f(t,y,z,v)−f(t,y′,z,v)|≤l|y−y′|, for t∈[0,T],y,y′∈Rk,z∈Rk×d,v∈Lλ2 . Moreover, we assume also that there exists a constant ϵ>0 0$]]> such that |ξ|+supt∈[0,T]|f(t,0,0,0)|+|R|T≤ϵ. For (Γ,Υ,Ψ,N)∈Ξ2 , in view of the assumptions made on ξ, f, and R, define the processes (Y,Z,V,M) as follows: Yt=E[ξ+∫0Tf(s,Γs,Υs,Ψs)ds+∫0TdRs|Ft]−∫0tf(s,Γs,Υs,Ψs)ds−∫0tdRs, and the local martingale E[ξ+∫0Tf(s,Γs,Υs,Ψs)ds+∫0TdRs|Ft]−Y0, which thanks to the martingale representation theorem (see Lemma 1), can be decomposed as follows: ∫0tZsdWs+∫0t∫UVs(e)πˆ(de,ds)+Mt, where Z, V, and M belong respectively to Lloc2(W) , Gloc(π) , and Mloc . Therefore, (Y,Z,V,M) is the unique solution of the GBSDE Yt=ξ+∫tTf(s,Γs,Υs,Ψs)ds+∫tTdRs−∫tTZsdWs−∫tT∫UVs(e)πˆ(de,ds)−∫tTdMs,t∈[0,T]. Moreover, from the conditions on ξ, f and R it is easy to prove that (Y,Z,V,M)∈Ξ2 .

As a by-product, we may define the mapping Φ:Ξ2→Ξ2 that associates (Γ,Υ,Ψ,N) with Φ((Γ,Υ,Ψ,N))=(Y,Z,V,M) . By standard arguments (see, e.g., the proof of [13, Thm. 55.1]) it can be shown that Φ is contractive on the Banach space Ξ2 endowed with the norm ‖(Y,Z,V,M)‖β=E{supt∈[0,T]eβt|Yt|2+∫0Teβt|Zt|2dt+∫0T∫Ueβt|Vt(e)|2λ(de)dt+[∫0.eβtdMt]T}12, for a suitably chosen constant β>0 0$]]>. Consequently, Φ has a fixed point (Y,Z,V,M)∈Ξ2 . Therefore, clearly, (Y,Z,V,M) is the unique solution of GBSDE (1) under the assumptions made so far.

Step 2. In this step, we will show how to dispense with assumption (22). We state and prove the following lemma.

Assume that (H1)p=2 , (H2) , (H3) , (Hgg) , (H5) , (H6) , and (23) hold. For given (Υ,Ψ)∈M2×L2 , there exists a unique quadruple of processes (Y,Z,V,M)∈Ξ2 such that Yt=ξ+∫tTf(s,Ys,Υs,Ψs)ds+∫tTdRs−∫tTZsdWs−∫tT∫UVs(e)πˆ(de,ds)−∫tTdMs.

For notational convenience, we set f(t,y)=f(t,y,Υt,Ψt) for each y∈Rk .

Uniqueness is proved by arguing as for uniqueness in Theorem 1. The result follows immediately. For the existence part, we follow the line of the proof of [14, Prop. 2.4]. Now, let us assume that (23) holds and define fn(t,y)=(ρn∗f(t,·))(y) , where ρn:R→R+ is a sequence of smooth functions with compact support that approximate the Dirac measure at 0 and satisfy ∫ρn(z)dz=1 . Moreover, they are defined such that ϖ satisfying ϖ(r)=supnsup|y|≤r∫Rγ(|y|)ρn(y−z)dz is finite for all r∈R+ . Note that f satisfies the following assumptions:

|f(t,y)|≤|f(t,0,0,0|+L(|Υs|+‖Ψs‖Lλ2)+γ(|y|) ,

E∫0T|f(t,0)|2dt<+∞ ,

(y−y′)(f(t,y)−f(t,y′))≤0 ,

y→f(t,y) is continuous for all t a.s.

 Thus, it is elementary to check that fn satisfies (i)–(iv) with the same constant L and ϖ instead of γ. However, we cannot apply step 1 of the proof since fn is not necessarily globally Lipschitz continuous in y but only locally Lipschitz. Hence, to overcome this problem, we add a truncation function Tp in fn . Indeed, define, for each p∈N , fn,p(t,y)=fn(t,Tp(y))such thatTp(y)=py|y|∨p. Notice that, for all n,p∈N , y→fn,p(t,y) is globally Lipschitz and satisfies the conditions of Step 1. Therefore, for all n,p∈N , according to what has already been proved in Step 1, there exists a unique solution (Yn,p,Zn,p,Vn,p,Mn,p) to GBSDE (25) associated with (ξ,fn,p+dR) . Furthermore, it follows from (23) and Lemma 2 with a=0 and q=t that there exists a universal constant C1>0 0$]]> such that, for all n,p∈N and t∈[0,T] , |Ytn,p|2+E(∫tT|Zsn,p|2ds+∫tT∫U|Vsn,p(e)|2λ(de)ds+∫tTd[Mn,p]s|Ft)≤C1E[|ξ|2+(∫tT|f(s,0,0,0)|ds)2+(∫tTd|R|s)2|Ft]≤C1ϵ2(2+T2):=r2. Hence, for any p>r r$]]>, the sequence (Yn,p,Zn,p,Vn,p,Mn,p) does not depend on p. Then we denote it by (Yn,Zn,Vn,Mn) , and it is a solution to GBSDE (25) associated with (ξ,fn+dR) . Moreover, now fn satisfies the conditions of Lemma 2 with a constant independent of n, and thus the sequence (Yn,Un,Zn,Vn,Mn) is uniformly bounded, that is, supn∈NE[∫0T|Ytn|2dt+(∫0Tfn(t,Ytn)dt)2+∫0T|Ztn|2dt+∫0T∫U|Vtn(e)|2λ(de)dt+[Mn]T]≤C. Let us set Utn=fn(t,Ytn) for t∈[0,T] . Using the previous uniform estimate of the sequence {(Yn,Zn,Vn,Un,)}n and the Hilbert structure of L2(Ω×[0,T])×M2×L2×H2 , we deduce that we can extract subsequences, still denoted {n} , that weakly converge to some process (Y,Z,V,U) in L2(Ω×[0,T])×M2×L2×H2 , where H2 denotes the set of Ft -progressively measurable Rk -valued processes (Ut)t∈[0,T] such that ‖U‖H2:={E[(∫0T|Ut|dt)2]}12<+∞. Now we deal with the convergence of the martingale Mn . By estimate (27) it follows that supn≥1E|MTn|2<∞ . Thus, there exists a subsequence, still denoted {n} , such that MTn converges weakly to some random variable MT in L2(Ω) . Let Mt denote the martingale with terminal value MT .

Next, following [14, Prop. 2.4], we deduce, by using the martingale representation theorem (see Lemma 1) and orthogonality that the following weak convergence hold for the martingales in L2(Ω) : for each t∈[0,T] , ∫tTZsndWs→∫tTZsdWs,∫tT∫UVsn(e)πˆ(de,ds)→∫tT∫UVs(e)πˆ(de,ds),andMtn→Mt. Therefore, taking weak limits in the approximating equation, we get that (Y,Z,V,M) satisfies the GBSDE Yt=ξ+∫tTUsds+∫tTdRs−∫tTZsdWs−∫tT∫UVs(e)πˆ(de,ds)−∫tTdMs. Finally, as in [14, Prop. 2.4], we can show that Ut=f(t,Yt) . This implies that (Y,Z,V,M) solves GBSDE (25) under the assumptions made so far in this step, which completes the proof of Lemma 3.

Step 3. In this step, we will show that assumption (Hgg) assumed so far can be weakened to (H4) . In fact, by the mean of truncation technique we will show that, for given (Υ,Ψ)∈M2×L2 and provided that (H1)p=2 – (H6) and (23) hold, GBSDE (25) has a solution in Ξ2 . The idea behind the proof is to approximate f by a sequence of functions fn satisfying assumption (Hgg) . Indeed, let θr be a smooth function such that 0≤θr≤1 and satisfies for r large enough: θr(y)={1for|y|≤r,0for|y|≥r+1. Let φr(t)=sup|y|≤r|f(t,y,0,0)−f(t,0,0,0)|∈L1([0,T]) and, for n∈N∗ , denote Tn(x)=xn|x|∨n . The approximation sequence fn is defined by fn(t,y,Υ,Ψ)=(f(t,y,Tn(Υt),Tn(Ψt))−f(t,0,Υt,Ψt))nφr+1(t)∨n+f(t,0,Υt,Ψt). We also define a sequence hn that truncates fn for |y|≥r+1 : hn(t,y,Υt,Ψt)=θr(y)(f(t,y,Tn(Υt),Tn(Ψt))−f(t,0,Υt,Ψt))nφr+1(t)∨n+f(t,0,Υt,Ψt). Following the same reasoning as in the proof of [4, Thm. 4.2], it can be shown that hn still satisfies the monotonicity condition (H3) but with a positive constant C(r,k,n) depending on r,k , and n. Then, the conditions of Lemma 3 of the previous step are fulfilled by the data (ξ,hn+dR) . Consequently, for each n∈N∗ , the GBSDE (25) associated with (ξ,hn+dR) , admits a unique solution (Yn,Zn,Vn,Mn)∈Ξ2 . Moreover, since yhn(t,y,Υ,Ψ)≤|y|‖f(t,0,0,0)‖∞+k|y|(|Υ|+‖Ψ‖Lλ2) , hn satisfies the condition of Lemma 2. Consequently, applying Lemma 2 with a=0 and q=t=0 , in view of the boundedness assumption (23), we get similarly as in (26) that, for each n∈N , the following estimates hold dP×dt -a.e.: |Ytn|≤randE(∫0T|Zsn|2ds+∫0T∫U|Vsn(e)|2λ(de)ds+[Mn]T)≤r2. As a by-product, (Yn,Zn,Vn,Mn) is a solution to the GBSDE (25) associated with the data (ξ,fn+dR) . Next, we show as in [9, Thm. 1] (see also [4, Thm. 4.2]) by using similar arguments that (Yn,Zn,Vn,Mn) is a Cauchy sequence in Ξ2 , and its limit is (Y,Z,V,M)∈Ξ2 .

Step 4. We now treat the general case. We want to get rid of the boundedness condition (23) used so far. To this end, a truncation procedure. Indeed, under assumptions (H1)p=2 – (H6) , we first set, for each n∈N∗ , ξn=Tn(ξ),fn(t,y)=fn(t,y,Υt,Ψt)=f(t,y)−f(t,0)+Tn(f(t,0)),Rtn=∫0t1{|R|s≤n}dRs. Then, according to the previous step, for each n∈N∗ , GBSDE (25) associated with (ξn,fn+dRn) has a unique solution (Yn,Zn,Vn,Mn)∈Ξ2 . Our goal now is to show that (Yn,Zn,Vn,Mn) is a Cauchy sequence in Ξ2 . Set (Y¯,Z¯,V¯,M¯)=(Ym−Yn,Zm−Zn,Vm−Vn,Mm−Mn) . Then (Y¯,Z¯,V¯,M¯) is solution to the GBSDE Y¯t=ξm−ξn+∫tTd(Rsm−Rsn)+∫tT(fm(s,Ysm)−fn(s,Ysn))ds−∫tTZ¯sdWs−∫tT∫UV¯s(e)πˆ(de,ds)−∫tTdM¯s,t∈[0,T]. Thanks to (H3) , (H5) , and (H6) , the generator of GBSDE (29) satisfies assumption (A) with ft≡Tm(f(t,0))−Tn(f(t,0)) and L=0 . Therefore, applying Lemma 2 with a=0 and q=t=0 yields, for all n,m∈N , E[supt∈[0,T]|Y¯t|2+∫0T|Z¯t|2dt+∫0T∫U|V¯t(e)|2λ(de)dt+[M¯]T]≤CE[|ξm−ξn|2+(∫0T|Tm(f(t,0))−Tn(f(t,0))|dt)2+(∫0Td|Rm−Rn|s)2]. Obviously, the right-hand side of (30) tends to 0 as n,m→∞ . Therefore, (Yn,Zn,Vn,Mn) is a Cauchy sequence in Ξ2 , and its limit (Y,Z,V,M)∈Ξ2 is an L2 -solution of GBSDE (25).

Step 5. In this step, we will finally complete the proof of the existence part of Theorem 1. To this end, we consider a Picard’s iteration procedure. Set (Y0,Z0,V0,M0)=(0,0,0,0) and define {(Ytn,Ztn,Vtn,Mtn)t∈[0,T]}n≥1 recursively in view of Step 4, for all n≥0 and t∈[0,T] , Ytn+1=ξ+∫tTf(s,Ysn+1,Zsn,Vsn)ds+∫tTdRs−∫tTZsn+1dWs−∫tT∫UVsn+1(e)πˆ(de,ds)−∫tTdMsn+1. From the previous step it follows that, under assumptions (H1)p=2 – (H6) , for each n≥0 , there exists a solution of GBSDE (31). Let us set δYn:=Yn+1−Yn , δZn:=Zn+1−Zn , δVn:=Vn+1−Vn , and δMn:=Mn+1−Mn . (δYn,δZn,δVn,δMn) solves the GBSDE δYtn=∫tT[f(s,Ysn+1,Zsn,Vsn)−f(s,Ysn,Zsn−1,Vsn−1)]ds−∫tTδZsndWs−∫tT∫UδVsn(e)πˆ(de,ds)−∫tTdδMsn,t∈[0,T]. By assumptions (H3) , (H5) , and (H6) we have that sgnˆ(δyn)(f(t,yn+1,zn,vn)−f(t,yn,zn−1,vn−1))=sgnˆ(δyn)[(f(t,yn+1,zn,vn)−f(t,yn,zn,vn))+(f(t,yn,zn,vn)−f(t,yn,zn−1,vn−1))]≤μ|δyn|+L|δzn−1|+L‖δvn−1‖Lλ2, which is assumption (A) with ft=L|δzn−1|+L‖δvn−1‖Lλ2 and L≡0 . Thus, it follows from Lemma 2 with a=q=t=0 and Hölder’s inequality that there exists a constant C>0 0$]]> such that E[supt∈[0,T]|δYtn|2+∫0T|δZsn|2ds+∫0T∫U|δVsn(e)|2λ(de)ds+[δMn]T]≤CE[∫0TL(|δZtn−1|+‖δVtn−1(e)‖Lλ2)dt]2≤CL2TE[∫0T(|δZtn−1|2+‖δVtn−1(e)‖Lλ22)dt]. Consequently, by induction we deduce that, for n≥2 , E[supt∈[0,T]|δYtn|2+∫0T|δZtn|2dt+∫0T∫U|δVtn(e)|2λ(de)dt+[δMn]T]≤cn−1E[∫0T|δZt1|2dt+∫0T∫U|δVt1(e)|2λ(de)dt], where c=CL2T . Let us first assume, for a sufficiently small T, that c<1 . Then, since the remaining term of the right-hand side of the last inequality is finite, we deduce that (Yn,Zn,Vn,Mn) is a Cauchy sequence in Ξ2 , and the limit process (Y,Z,V,M) is a solution to GBSDE (1) in Ξ2 .

For the general case, it suffices to subdivide the interval time [0,T] into a finite number of small intervals, and using the standard arguments, we can prove the existence of a solution (Y,Z,V,M) of GBSDE (1) in Ξ2 on the whole interval [0,T] . This completes the proof of this step and thus of the whole proof of Theorem 1.  □

 Case <inline-formula id="j_vmsta73_ineq_458"><alternatives> <mml:math><mml:mi mathvariant="italic">p</mml:mi><mml:mo stretchy="false">≥</mml:mo><mml:mn>2</mml:mn></mml:math> <tex-math><![CDATA[$p\ge 2$]]></tex-math></alternatives></inline-formula>

In this subsection, we study the issue of existence and uniqueness of Lp -solutions of GBSDE (1) in the case p≥2 . Let us first give a priori estimates for the solution and their variations induced by a variation of the data.

Let assumption (A) hold, and let (Y,Z,V,M) be a solution of GBSDE (1). Let us assume moreover that E[|ξ|p+(∫0Tftdt)p+|R|Tp]<+∞. If Y∈Sp , then there exists a constant Cp>0 0$]]>, depending only on p and T, such that, for every a≥μ+2L2 , E[supt∈[0,T]eapt|Yt|p+(∫0Te2at|Zt|2dt)p/2+(∫0T∫Ue2at|Vt(e)|2λ(de)dt)p/2+eapT[M]Tp/2]≤CE[eapT|ξ|p+(∫0Teatftdt)p+(∫0Teatd|R|t)p].

The proof is divided into two steps. By an already used argument (see Lemma (2)) we can assume w.l.o.g. that μ+2L2≤0 and take a=0 .

Step 1. First, we show that E[(∫0T|Zs|2ds)p2+(∫0T∫U|Vs(e)|2λ(de)ds)p2+(∫0Td[M]s)p2]≤C(p,T)[Esupt∈[0,T]|Yt|p+(∫0Tfsds)p+(∫0Td|R|s)p]. Indeed, define the sequence of stopping times τn for n∈N : τn=inf{t>0;∫0t|Zs|2ds+∫0t∫U|Vs(e)|2λde(ds)+[M]t>n}∧T. 0;{\int _{0}^{t}}|Z_{s}{|}^{2}ds+{\int _{0}^{t}}\int _{U}{\big|V_{s}(e)\big|}^{2}\lambda de(ds)+[M]_{t}>n\Bigg\}\wedge T.\]]]> By Itô’s formula on |Yt|2 , |Y0|2+∫0τn|Zs|2ds+∫0τn∫U|Vs(e)|2π(de,ds)+∫0τnd[M]s=|Yτn|2+2∫0τnYsf(s,Ys,Zs,Vs)ds+2∫0τnYs−dRs−2∫0τnYsZsdWs−2∫0τn∫UYs−Vs(e)πˆ(de,ds)−2∫0τnYs−dMs. But from assumption (A) , combined with the inequality 2ab≤1εa2+εb2 for ε>0 0$]]>, since μ+2L2<0 , we have 2Ysf(s,Ys,Zs,Vs)≤2μ|Ys|2+2|Ys||fs|+2L|Ys||Zs|+2L|Ys|‖Vs(e)‖Lλ2≤(1ε+2(μ+L2))|Ys|+2|Ys|fs+12|Zs|2+ε‖Vs(e)‖Lλ22≤1ε|Ys|2+2|Ys|fs+12|Zs|2+ε‖Vs(e)‖Lλ22. Thus, since τn≤T , we deduce that 12∫0τn|Zs|2ds+∫0τn∫U|Vs(e)|2π(de,ds)+∫0τnd[M]s≤supt∈[0,T]|Yt|2+1ε∫0T|Ys|2ds+2supt∈[0,T]|Yt|∫0Tfsds+2supt∈[0,T]|Yt|∫0Td|R|s+ε∫0τn∫U|Vs(e)|2λ(de)ds+2|∫0τnYsZsdWs|+2|∫0τn∫UYs−Vs(e)2πˆ(de,ds)|+2|∫0τnYs−dMs|≤(3+Tϵ)supt∈[0,T]|Yt|2+(∫0Tfsds)2+(∫0Td|R|s)2+ε∫0τn∫U|Vs(e)|2λ(de)ds+|∫0τnYsZsdWs|+|∫0τn∫UYs−Vs(e)2πˆ(de,ds)|+|∫0τnYs−dMs|. It follows that there exists a constant cp>0 0$]]>, depending only on p, such that (∫0τn|Zs|2ds)p2+(∫0τn∫U|Vs(e)|2π(de,ds))p2+(∫0τnd[M]s)p2≤cp[(3+Tϵ)p2supt∈[0,T]|Yt|p+(∫0Tfsds)p+(∫0τnd|R|s)p+ϵp2(∫0τn∫U|Vs(e)|2λ(de)ds)p2+|∫0τnYsZsdWs|p2+|∫0τn∫UYs−Vs(e)2πˆ(de,ds)|p2+|∫0τnYs−dMs|p2]. Since p2≥1 , we can apply the BDG inequality to obtain cpE|∫0τnYsZsdWs|p2≤dpE[(∫0τn|Ys|2|Zs|2ds)p4]≤dp24E(supt∈[0,T]|Yt|p)+12E(∫0τn|Zs|2ds)p2, cpE|∫0τnYs−dMs|p2≤dpE[(∫0τn|Ys−|2d[M]s)p4]≤dp24E(supt∈[0,T]|Yt|p)+12E[M]τnp2, and cpE|∫0T∫UYs−Vs(e)πˆ(de,ds)|p2≤dpE[(∫0τn|Ys|2|Vs(e)|2π(de,ds))p4]≤dp24E(supt∈[0,T]|Yt|p)+12E(∫0τn|Vs(e)|2π(de,ds))p2. Plugging estimates (37)–(39) into (36) and then taking the expectation, we get 12E[(∫0τn|Zs|2ds)p2+(∫0τn∫U|Vs(e)|2π(de,ds))p2+(∫0τnd[M]s)p2]≤C(p,T,ϵ)Esupt∈[0,T]|Yt|p+cpE(∫0Tfsds)p+cpE(∫0τnd|R|s)p+cpϵp2E[(∫0τn∫U|Vs(e)|2λ(de)ds)p2]. By [11, Section 4] or [6, Lemma 2.1] we have that, for some constant ηp>0 0$]]>, E(∫0τn∫U|Vs(e)|2λ(de)ds)p2≤ηpE(∫0τn∫U|Vs(e)|2π(de,ds))p2. Thus, choosing ϵ small enough and depending only on p, we deduce that E[(∫0τn|Zs|2ds)p2+(∫0τn∫U|Vs(e)|2λ(de)ds)p2+(∫0τnd[M]s)p2]≤C˜(p,T)Esupt∈[0,T]|Yt|p+C˜pE[(∫0Tfsds)p+(∫0Td|R|s)p]. Finally, letting n to +∞ and using Fatou’s lemma, (34) follows.

Step 2. Since p≥2 , we can apply Itô’s formula with the C2 function |y|p to |Yt|p . Note that ∂θ∂yi(y)=pyi|y|p−2,∂2θ∂yi∂yj(y)=p|y|p−2δi,j+p(p−2)yiyj|y|p−4, where δi,j is the Kronecker delta. Thus, for every t∈[0,T] , we have |Yt|p=|ξ|p+p∫tTYs−|Ys−|p−2dRs+p∫tTYs|Ys|p−2f(s,Ys,Zs,Vs)ds−p∫tTYs−|Ys−|p−2dMs−p∫tTYs|Ys|p−2ZsdWs−p∫tT∫U(Ys−|Ys−|p−2Vs(e))πˆ(de,ds)−12∫tTTrace(D2θ(Ys)ZsZst)ds−∫tT∫U(|Ys−+Vs(e)|p−|Ys−|p−pYs−|Ys−|p−2Vs(e))π(de,ds)−ℵt, where ℵt=12∫tT∑1≤i,j≤d∂2θ∂yi∂yj(Ys)d[Mi,Mj]sc+∑t<s≤T(|Ys−+ΔMs|p−|Ys−|p−pYs−|Ys−|p−2ΔMs). Following arguments from [9, Prop. 2], we have that Trace(D2θ(y)zzt)≥p|y|p−2|z|2, ℵt≥αp∫tT|Ys−|d[M]s, and −∫tT∫U(|Ys−+Vs(e)|p−|Ys−|p−pYs−|Ys−|p−2Vs(e))π(de,ds)≤−p(p−1)31−p∫tT|Ys−|p−2|Vs(e)|2π(de,ds), where αp=min(p2,p(p−1)31−p) .

Consequently, in view of estimates (42), (43), and (44), Eq. (41) becomes |Yt|p+αp∫tT|Ys|p−2|Z|s2ds+αp∫tT∫U|Ys|p−2|Vs(e)|2π(de,ds)+αp∫tT|Ys|p−2d[M]sc+αp∑t<s≤T|Ys|p−2|ΔMs|2≤|ξ|p+p∫tTYs−|Ys−|p−2dRs+p∫tTYs|Ys|p−2f(s,Ys,Zs,Vs)ds−p∫tTYs−|Ys−|p−2dMs−p∫tTYs|Ys|p−2ZsdWs−p∫tT∫UYs−|Ys−|p−2Vs(e)πˆ(de,ds). But from assumption (A) and the fact that μ≤−2L2≤0 (since μ+2L2≤0 ) we deduce by using the inequality ab≤12ϵa2+ϵ2b2 that Ysf(s,Ys,Zs,Vs)≤L2ϵ|Ys|2+|Ys|fs+ϵ2|Zs|2+ϵ2∫U|Vs(e)|2λ(de). Choosing ϵ=αpp , we obtain in view of the last inequality that |Yt|p+αp2∫tT|Ys|p−2|Z|s2ds+αp∫tT∫U|Ys|p−2|Vs(e)|2π(de,ds)+αp∫tT|Ys|p−2d[M]s≤|ξ|p+pL2αp∫tT|Ys|pds+p∫tT|Ys|p−1fsds+p∫tT|Ys−|p−1dRs+αp2∫tT|Ys|p−2‖Vs(e)‖Lλ22ds−p∫tTYs|Ys|p−2ZsdWs−p∫tT∫UYs−|Ys−|p−2Vs(e)πˆ(de,ds)−p∫tTYs−|Ys−|p−2dMs. Let us set X=|ξ|p+pL2αp∫tT|Ys|pds+p∫tT|Ys|p−1|fs|ds+p∫tT|Ys−|p−1dRs , Υt=∫0tYs|Ys|p−2ZsdWs , Θt=∫0t∫UYs−|Ys−|p−2Vs(e)πˆ(de,ds) , and Γt=∫0tYs−|Ys−|p−2dMs .

It follows from the BDG inequality that Υt , Θt , and Γt are uniformly integrable martingales. Indeed, by Young’s inequality we have E([Υ]T12)≤E[supt∈[0,T]|Yt|p−1(∫0T|Zs|2ds)12]≤p−1pE(supt∈[0,T]|Yt|p)+1pE(∫0T|Zs|2ds)p2, E([Θ]T12)≤E[supt∈[0,T]|Yt|p−1(∫0T|Vs(e)|2π(de,ds))12]≤p−1pE(supt∈[0,T]|Yt|p)+1pE(∫0T‖Vs(e)‖Lλ22ds)p2, and E([Γ]T12)≤E[supt∈[0,T]|Yt|p−1[M]T12]≤p−1pE(supt∈[0,T]|Yt|p)+1pE[M]Tp2. The claim holds since the last terms of (48), (49), and (50) are finite. This is due to the fact that Y∈Sp , which implies by the first step of the proof that Z∈Mp , V∈Lp , and M∈Mp . Moreover, we have E∫tT∫U|Ys|p−2|Vs(e)|2π(de,ds)=E∫tT|Ys|p−2‖Vs(e)‖Lλ22ds. Hence, in view of (51), taking the expectation in (47) yields αp2E∫tT|Ys|p−2|Z|s2ds+αp2E∫tT|Ys|p−2‖Vs(e)‖Lλ22ds+αpE∫tT|Ys|p−2d[M]s≤E[X].

Furthermore, coming back to (47), we deduce in view of (52) that Esups∈[t,T]|Ys|p≤2EX+pE(sups∈[t,T]|∫sT∫UYu−|Yu−|p−2Vu(e)πˆ(de,du)|)+pE(sups∈[t,T]|∫sTYu|Yu|p−2ZudWu|)+pE(sups∈[t,T]|∫sTYu−|Yu−|p−2dMu|). The BDG inequality implies that pE(sups∈[t,T]|∫sTYu|Yu|p−2ZudWu|)≤dpE[(∫tT|Ys|2p−2|Zs|2ds)12]≤14Esups∈[t,T]|Ys|p+4dp2E(∫tT|Ys|p−2|Zs|2du), pE(sups∈[t,T]|∫sT∫UYu−|Yu−|p−2Vu(e)πˆ(de,du)|)≤dpE(∫tT∫U|Yu|2p−2|Vs(e)|2π(de,ds))12≤14Esups∈[t,T]|Ys|p+4dp2E(∫tT|Yu|p−2‖V(e)u‖Lλ22du), and pE(sups∈[t,T]|∫sTYu−|Yu−|p−2dMu|)≤dpE(|Ys|2p−2d[M]s)12≤14Esups∈[t,T]|Ys|p+4dp2E(∫tT|Ys|p−2d[M]s). Thus, combining estimates (54)–(56) with (52), we deduce that Esups∈[t,T]|Ys|p≤CpE[X]. But, applying Young’s inequality, we get pCpE∫tT|Ys|p−1|fs|ds≤pCpE(sups∈[t,T]|Yt|p−1∫tTfsds)≤16Esups∈[t,T]|Yt|p+dp′E(∫tTfsds)p, and pCpE∫tT|Ys−|p−1dRs≤16Esups∈[t,T]|Yt|p+dp″E(∫tTd|R|s)p. Consequently, rearranging (57) in view of the two last estimates implies Esups∈[t,T]|Ys|p≤Cp′E[|ξ|p+(∫tTfsds)p+(∫tTd|R|s)p]+Cp″∫tTEsupu∈[s,T]|Yu|pds,t∈[0,T]. Finally, using Gronwall’s lemma, we deduce that Esupt∈[0,T]|Yt|p≤Cp′eCp″TE[|ξ|p+(∫0Tfsds)p+(∫0Td|R|s)p]. This, combined with (34), ends the proof.  □

Let (ξ,f,R) and (ξ′,f′,R′) be two sets of data, each satisfying assumptions (H1) – (H6) . Let (Y,Z,V,M) and (Y′,Z′,V′,M′) denote respectively an Lp -solution of GRBSDE (1) with data (ξ,f,R) and (ξ′,f′,R′) . Define (Y¯,Z¯,V¯,M¯,ξ¯,f¯,R¯)=(Y−Y′,Z−Z′,V−V′,M−M′,ξ−ξ′,f−f′,R−R′). Then there exists a constant C>0 0$]]>, depending on p and T, such that, for every a≥μ+2L2 , E[supt∈[0,T]eapt|Y¯t|p+(∫0Te2at|Z¯t|2dt)p2+(∫0T∫Ue2at|V¯t|2λ(de)dt)p2+eapT[M¯]Tp2]≤CE[eapT|ξ¯|p+(∫0Teat|f¯(s,Yt′,Zt′,Vt′)|dt)p+(∫0Teatd|R¯|t)p].

By an already used change-of-variable argument we may assume that a=0 . Obviously, (Y¯,Z¯,V¯,M¯) solves the following GBSDE in Ξp : Y¯t=ξ¯+∫tT(f(s,Ys,Zs,Vs)−f′(s,Ys′,Zs′,Vs′))ds+∫tTdR¯s−∫tTZ¯sdWs−∫tT∫UV¯s(e)πˆ(de,ds)−∫tTdM¯s,t∈[0,T]. It follows from (H3) , (H5) , and (H6) that sgnˆ(y¯)(f(t,y,z,v)−f′(t,y′,z′,v′))=sgnˆ(y¯)(f(t,y,z,v)−f(t,y′,z′,v′))+sgnˆ(y¯)f¯(t,y′,z′,v′)=sgnˆ(y¯)[f(t,y,z,v)−f(t,y′,z,v)+f(t,y′,z,v)−f(t,y′,z′,v)+f(t,y′,z′,v)−f(t,y′,z′,v′)]+sgnˆ(y¯)f¯(t,y′,z′,v′)≤|f¯(t,y′,z′,v′)|+μ|y|+L|z|+L‖v‖Lλ2, which means that assumption (A) is satisfied for the generator of GBSDE (60), with ft≡|f¯(t,y′,z′,v′)| . Thus, by Lemma 4 the desired estimate follows, which ends the proof of Lemma 5.  □

Now we are able to give the main result of this subsection, the existence and uniqueness of an Lp -solution of GBSDE (1) in the case p≥2 .

Let p≥2 and assume that (H1) – (H6) hold. Then, there exists a unique Lp -solution (Y,Z,V,M) for the GBSDE (1).

Uniqueness follows immediately from Lemma 5. Now we deal with the existence. Set Tn(x)=xn|x|∨n for n∈N∗ and define ξn , fn , and Rn as follows: ξn=Tn(ξ),fn(t,y,z,v)=f(t,y,z,v)−f(t,0,0,0)+Tn(f(t,0,0,0)),Rtn=∫0t1{|R|s≤n}dRs. Let (Yn,Zn,Vn,Mn) be a solution of GBSDE (1) associated with (ξn,fn+dRn) . Hence, by Theorem 1, for every n∈N , there exists a unique solution to (Yn,Zn,Vn,Mn)∈Ξ2 of GBSDE (1) associated with (ξn,fn+dRn) , but in fact also in Ξp , p≥2 , according to Lemma 4. Our goal now is to show that (Yn,Zn,Vn,Mn) is a Cauchy sequence in Ξp . For m≥n , applying Lemma 5 yields E[supt∈[0,T]|Ytm−Ytn|2+(∫0T|Ztm−Ztn|2dt)p2+(∫0T∫U|Vtm(e)−Vtn(e)|2λ(de)dt)p2+[Mm−Mn]Tp2]≤CE[|ξm−ξn|p+(∫0T|Tm(f(t,0,0,0))−Tn(f(t,0,0,0))|dt)p+(∫0Td|Rm−Rn|s)p]. Therefore, letting n and m to infinity, we conclude that (Yn,Zn,Vn,Mn) is a Cauchy sequence in Ξp and its limit (Y,Z,V,M)∈Ξp is a solution of GBSDE (1) associated with (ξ,f+dR) , which ends the proof.  □

 Comparison theorem

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 dWt , respectively. Let γ be a predictable process defined on [0,T]×Ω×R integrable w.r.t.  πˆ(de,ds) . For any 0≤t≤s≤T , let E be the solution of dEt,s=Et,s−[βsdWs+∫Uγs(e)πˆ(de,ds)],Et,t=1, and let Γ be the solution of dΓt,s=Γt,s−[αsds+βsdWs+∫Uγs(e)πˆ(de,ds)],Γt,t=1. Of course, Γt,s=exp(∫tsαrdr)Et,s , and Et,s=exp(∫tsβrdWr−12∫tsβr2dr)∏t<r≤s(1+γr(ΔXr))e−γr(ΔXr), with Xt=∫0t∫Uuπ(du,ds) .

Note that, classically, if γt(e)≥−1,dP⊗ds⊗dλ(e) -a.s., then Γt,.≥0 a.s. (see [16, Prop. 3.1]).

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

For each (y,z,v,v′)∈R×Rd×(Lλ2)2 , there exists a predictable process κ=κy,z,v,v′:Ω×[0,T]×U→R satisfying:

−1≤κty,z,v,v′(e) ;

|κty,z,v,v′(e)|≤ϑ(e) , where ϑ∈Lλ2 is such that

 f(t,y,z,v)−f(t,y,z,v′)≤∫U(v(e)−v′(e))κty,z,v,v′(e)λ(de), P⊗Leb⊗λ -a.e.

Notice that (H6′) 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.

Assume that |β| is bounded and α is bounded from above. Suppose also that, dP⊗dt⊗λ(de) -a.s., −1≤γt(e), and |γt(e)|≤ϑ(e),whereϑ∈Lλ2. Let (ft)0≤t≤T be a real-valued progressively measurable process, and let (Y,Z,V,M) be the solution of the linear GBSDE Yt=ξ+∫tT[fs+αsYs+βsZs+∫Uγs(u)Vs(e)λ(de)]ds+∫tTdRs−∫tT∫UVs(e)πˆ(de,ds)−∫tTZsdWs−∫tTdMs. Then, Esups∈[t,T]|Γt,s|p<+∞ , and if E[|ξ|p+(∫0T|fs|ds)p+|R|Tp]<+∞, then the solution (Y,Z,V,M) belongs to Ξp .

Furthermore, the process (Yt) satisfies Yt=E[ξΓt,T+∫tTΓt,sfsds+∫tTΓt,s−dRs|Ft],0≤t≤T,a.s.

We first show that Esups∈[t,T]|Γt,s|p<+∞ . Indeed, by (63), as mentioned previously, it follows that Γt,.≥0 . Combining this with (64), using the fact that |β| is bounded and α is bounded from above, and applying [16, Prop. A.1] yield that Γ is p-integrable, that is, E|Γt,T|p<+∞ . Hence, using Doob’s inequality, we have that Esups∈[t,T]|Γt,s|p≤Cpsups∈[t,T]E|Γt,s|p≤CpE|Γt,T|p<+∞, as desired.

Next, let us show that (Y,Z,V,M) belongs to Ξp . Clearly, thanks to the assumptions made on α, β, and γ and the fact (H6′) implies (H6) , we can easily see that the generator of the linear GBSDE (65) satisfies assumptions (H3) , (H5) , and (H6) . Thus, in view of this and (66), applying Lemma (4) yields the claim.

It remains to show that (Yt) satisfies (67). Indeed, by the Itô product formula we obtain d(YsΓt,s)=Γt,s−dYs+Ys−dΓt,s+d[Γt,.,Y]s=Γt,s−(−fs−αsYs−βsZs−∫Uγs(u)Vs(u)λ(de))ds−Γt,s−dRs+Γt,s−∫UVs(e)πˆ(de,ds)+Γt,s−ZsdWs+Γt,s−dMs+Ys−Γt,s−(αsds+βsdWs+∫Uγs(u)πˆ(de,ds))+Γt,s−βsZsds+Γt,s−∫UVs(e)γs(u)π(de,ds)=−Γt,sfsds−Γt,s−dRs+dNs, with dNs=Γt,s−∫U(Vs(e)+Ys−γs(e)+Vs(e)γs(e))πˆ(de,ds)+Γt,s−(Zs+Ysβs)dWs+Γt,s−dMs. Integrating between t and T yields ξΓt,T−Yt=−∫tTΓt,sfsds−∫tTΓt,s−dRs+∫tTdNsa.s. In view of the boundedness assumptions made on the coefficients β and γ, combined with estimate (68) and the fact that (Y,Z,V,M)∈Ξp , it follows that the local martingale N is a uniformly integrable martingale. Therefore, taking the conditional expectation w.r.t. Ft in (69), we obtain Yt=E[ξΓt,T+∫tTΓt,sfsds+∫tTΓt,s−dRs|Ft], as desired. This ends the proof.  □

We consider two sets of data (ξ1,f1+dR1) and (ξ2,f2+dR2) such that ξ1 , ξ2 , R1 , and R2 satisfy (H1) . Moreover, we assume that f1 and f2 satisfy, respectively, (H1) – (H6) and (H1) – (H5) , (H6′) . Let (Y1,Z1,V1,M1) and (Y2,Z2,V2,M2) be respectively solutions of GBSDEs (1) associated with (ξ1,f1+dR1) and (ξ2,f2+dR2) in some space Ξp with p≥2 . If ξ1≤ξ2 , f1(t,Yt1,Zt1,Vt1)≤f2(t,Yt1,Zt1,Vt1) , and for a.e. t, dR1≤dR2 , then a.s. Yt1≤Yt2 for any t∈[0,T] .

Put Y‾=Y2−Y1,Z‾=Z2−Z1,V‾=V2−V1,M‾=M2−M1,R‾=R2−R1. Then (Y‾,Z‾,V‾,M‾) satisfies Y‾t=ξ‾+∫tThsds+∫tTdR‾s−∫tT∫Uψ‾s(u)πˆ(de,ds)−∫tTZ‾sdWs−∫tTdM‾s, where hs=f2(Ys2,Zs2,ψs2)−f1(Ys1,Zs1,ψs1). Now we define fs=f2(Ys1,Zs1,ψs1)−f1(Ys1,Zs1,ψs1),αs=f2(Ys2,Zs1,ψs1)−f2(Ys1,Zs1,ψs1)Y‾s1Y‾s≠0,βs=f2(Ys2,Zs2,ψs1)−f2(Ys2,Zs1,ψs1)Z‾s1Z‾s≠0. Then hs=fs+αsY‾s+βsZ‾s+f2(Ys2,Zs2,Vs2)−f2(Ys2,Zs2,Vs1)≥fs+αsY‾s+βsZ‾s+∫UκsYs2,Zs2,Vs1,Vs2V‾s(u)λ(de), since f2 satisfies (H6′) . Moreover, since f2 is Lipschitz continuous w.r.t. z, |β| is bounded by L, whereas, by Assumption (H3) , α is bounded from above. Moreover, the process κsYs2,Zs2,Vs1,Vs2 is controlled by ϑ∈Lλ2 . Note that, since −1≤κty,z,ψ,ϕ(e) , it follows that Γt,.≥0 a.s. Furthermore, in view of the above, we have from Lemma 6 that Γt,.∈Sp .

Now applying Itô’s formula to Y¯sΓt,s for s∈[t,T] and then using inequality (70) together with the non negativity of Γ, we can derive, by doing the same computations as in the proof of Lemma 6, that −d(Y¯sΓt,s)≥Γt,sfsds+Γt,s−dR¯s−dNs, where N is a local martingale. Next, applying Lemma 5 yields that (Y¯,Z¯,V¯,M¯) belongs to Ξp . Since Γt,.∈Sp , this, combined with the boundedness of β and κty,z,ψ,ϕ(e) , implies that N is in fact a martingale.

Therefore, integrating between t and T in (71) and then taking the conditional expectation w.r.t. Ft , we deduce Y‾t≥E[Γt,Tξ‾+∫tTΓt,sfsds+∫tTΓt,s−dR‾s|Ft],t∈[0,T],a.s. To conclude, recall that Γt,s≥0 a.s. and, by assumptions, ξ‾≥0 , fs≥0 , and ∫tTdR‾s≥0 . Consequently, it follows that, for all t∈[0,T] ,  Y‾t≥0 a.s. Since Y1 and Y2 are càdlàg processes, we obtain that Yt1≤Yt2 a.s., and the conclusion follows.  □

Notice that assumptions (H1) – (H6) made on f1 are imposed only to ensure the existence of a solution (Y1,Z1,V1,M1) . The following corollary, which follows immediately from Proposition 1, gives again a uniqueness result for GBSDE (1) in Ξp in dimension 1.

Let p≥2 and assume (H1) – (H5) and (H6′) . Then there exists at most one solution (Y,Z,V,M) to GBSDE (1) in Ξp .

 Generalized BSDEs with random terminal time

In this section, we study the issue of existence and uniqueness of Lp(p≥2) -solutions of GBSDEs with random terminal time. We follow the approach in [14, Section 4]. Let τ be an 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≥2 :

∀r>0 0$]]>, ∀n∈N , the mapping t∈[0,T]→sup|y|≤r|f(t,y,0,0)−f(t,0,0,0)| belongs to L1(Ω×(0,n)) .

For some ρ∈R such that ρ>ν:=μ+2pL2αp \nu :=\mu +\frac{2p{L}^{2}}{\alpha _{p}}$]]>, where αp=min(p2,p(p−1)31−p) , E[eρpτ|ξ|p+(∫0τeρτ|f(s,0,0,0)|ds)p+(∫0τeρτd|R|s)p]<∞.

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

(H7) ξ is Fτ -measurable, and E[(∫0τeρτ|f(t,ξt,ηt,γt)|ds)p]<+∞ , where ξt=E(ξ|Ft) and (η,γ,N) are given by the martingale representation ξ=E(ξ)+∫0+∞ηsdWs+∫0+∞∫Uγs(e)πˆ(de,ds)+Nτ, with N orthogonal to W and π˜ . Moreover, the following holds: E[(∫0+∞|ηs|2ds)p2+(∫0+∞∫U|γs(e)|2λ(de)ds)p2+[N]τp2]<+∞.

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

We say that a quadruple (Y,Z,V,M)∈S×H(0,T)×P×Mloc with values in Rk×Rk×d×Rk×Rk is a solution of GBSDE (1) with random terminal time τ and data (ξ,f+dR) if

on {t≥τ} , Yt=ξ and Zt=Vt=Mt=0 , P -a.s.,

t→f(t,Yt,Zt,Vt)1{t≤τ}∈L1(0,+∞) , Z∈Lloc2(W) , V∈Gloc(π) , and

P -a.s., for all t∈[0,T] , Yt∧τ=YT∧τ+∫t∧τT∧τf(s,Ys,Zs,Vs)ds+∫t∧τT∧τdRs−∫t∧τT∧τZsdWs−∫t∧τT∧τ∫UVs(e)πˆ(de,ds)−∫t∧τT∧τdMs.

Furthermore, a solution is said to be Lp if we have E[epρ(t∧τ)|Yt∧τ|p+∫0T∧τepρs|Ys|pds+∫0T∧τepρs|Ys|p−2|Zs|2ds+∫0T∧τepρs|Ys|p−2‖Vs‖Lλ22ds+∫0T∧τepρs|Ys|p−2d[M]s]<+∞.

Assume (H1′) , (H2) , (H3) , (H4′) , (H5) , (H6) , and (H7) . Then, there exists at most one solution to GBSDE (1) that satisfies estimate (74).

Assume that there exist two solutions (Y,Z,V,M) and (Y′,Z′,V′,M′) of GBSDE (73) that satisfy estimate (74). Set (Y¯,Z¯,V¯,M¯)=(Y−Y′,Z−Z′,V−V′,M−M′) .

Applying Itô’s formula, as in step 2 of Proposition 4, to epρs|Y¯s|p over the interval [t∧τ,T∧τ] , we obtain an analogue of (41) epρ(t∧τ)|Y¯t∧τ|p=epρ(T∧τ)|Y¯T∧τ|p+p∫t∧τT∧τepρs[Y¯s|Y¯s|p−2(f(s,Ys,Zs,Vs)−f(s,Ys′,Zs′,Vs′))−ρ|Y¯s|p]ds−p∫t∧τT∧τepρsY¯s−|Y¯s−|p−2dM¯s−p∫t∧τT∧τepρsY¯s|Y¯s|p−2Z¯sdWs−p∫t∧τT∧τ∫UepρsY¯s−|Y¯s−|p−2V¯s(e)πˆ(de,ds)−12∫t∧τT∧τepρsTrace(D2θ(Y¯s)Z¯sZ¯st)ds−∫t∧τT∧τ∫Uepρs(|Y¯s−+Y¯s(e)|p−|Y¯s−|p−pY¯s−|Y¯s−|p−2Y¯s(e))π(de,ds)−ℵt, where ℵt=12∫t∧τT∧τepρs∑1≤i,j≤d∂2θ∂yi∂yj(Ys)d[M¯i,M¯j]sc+∑t∧τ<s≤T∧τepρs(|Y¯s−+ΔM¯s|p−|Y¯s−|p−pYs−|Y¯s−|p−2ΔM¯s). The following estimates, which are analogues of (43) and (44), hold: ℵt≥αp∫t∧τT∧τepρs|Y¯s−|d[M¯]s, and −∫t∧τT∧τ∫Uepρs(|Y¯s−+V¯s(e)|p−|Y¯s−|p−pY¯s−|Y¯s−|p−2Y¯s(e))π(de,ds)≤−p(p−1)31−p∫t∧τT∧τepρs|Y¯s−|p−2|Y¯s(e)|2π(de,ds), where αp=min(p2,p(p−1)31−p) .

Therefore, rearranging (75), in view of (42), (77), and (78) yields epρ(t∧τ)|Y¯t∧τ|p+αp∫t∧τT∧τepρs|Y¯s|p−2|Z¯s|2ds+αp∫t∧τT∧τepρs∫U|Y¯s|p−2|V¯s(e)|2π(de,ds)+αp∫t∧τT∧τepρs|Y¯s−|d[M¯]s≤epρ(T∧τ)|Y¯T∧τ|p+p∫t∧τT∧τeρs[Y¯s|Y¯s|p−2(f(s,Ys,Zs,Vs)−f(s,Ys′,Zs′,Vs′))−ρ|Y¯s|p]ds−p∫t∧τT∧τeρsY¯s−|Y¯s−|p−2dM¯s−p∫t∧τT∧τeρsY¯s|Y¯s|p−2Z¯sdWs−p∫t∧τT∧τ∫UeρsY¯s−|Y¯s−|p−2V¯s(e)πˆ(de,ds). But from the assumptions on f (using (46) with ϵ=αpp ) and Young’s inequality we have that Y¯s|Y¯s|p−2(f(s,Ys,Zs,Vs)−f(s,Ys′,Zs′,Vs′))−ρ|Y¯s|p≤(μ+2pL2αp−ρ)|Y¯s|p+αpp|Y¯s|p−2|Z¯s|2+αpp|Y¯s|p−2‖V¯s‖Lλ22≤αpp|Y¯s|p−2|Z¯s|2+αpp|Y¯s|p−2‖V¯s‖Lλ22. Furthermore, observe that by the integrability conditions on the solution all the local martingales appearing in (79) are uniformly integrable. Moreover, the following holds: E∫t∧τT∧τepρs∫U|Ys|p−2|Vs(e)|2π(de,ds)=E∫t∧τT∧τepρs∫U|Ys|p−2|Vs(e)|2λ(de)ds. Thus, taking the expectation in (79), we obtain, in view of the above, that Eepρ(t∧τ)|Y¯t∧τ|p≤Eepρ(T∧τ)|Y¯T∧τ|p. Note that the same result holds with ρ replaced by ρ′ , with μ+p2L2αp2<ρ′<ρ . Therefore, we have, for any 0≤t≤T , Eepρ′(t∧τ)|Y¯t∧τ|p≤ep(ρ′−ρ)TEepρ(T∧τ)|Y¯T∧τ|p. Consequently, letting T→+∞ , we deduce in view of estimate (74) that Y¯t=0 .

Since (Y,Z,V,M) and (Y′,Z′,V′,M′) satisfy GBSDE (73) with Y=Y′ , then by the uniqueness of the Doob–Meyer decomposition of semimartingales it follows that (Z,V,M)=(Z′,V′,M′) , whence the uniqueness of the solution of (73).  □

Assume that (H1′) , (H2) , (H3) , (H4′) , (H5) , (H6) , and (H7) are in force. Then, GBSDE (1) has a solution satisfying E[supt≥0epρ(t∧τ)|Yt∧τ|p+epρ(t∧τ)|Yt∧τ|p+∫0T∧τepρs|Ys|pds+∫0T∧τepρs|Ys|p−2|Zs|2ds+∫0T∧τepρs|Ys|p−2‖Vs‖Lλ22ds+∫0T∧τepρs|Ys−|p−2d[M]s]≤CE[epρτ|ξ|p+(∫0τeρτ|f(s,0,0,0)|ds)p+(∫0τeρτd|R|s)p]. Moreover, E[(∫0τe2ρs|Zs|2ds)p2+(∫0τ∫Ue2ρs|Vs(e)|2λ(de)ds)p2+(∫0τe2ρsd[M]s)p2]≤CE[epρτ|ξ|p+(∫0τeρτ|f(s,0,0,0)|ds)p+(∫0τeρτd|R|s)p], for some constant C>0 0$]]> depending only on p, L, and μ.

We follow the line of the argument of [14, Thm. 4.1]. For each n∈N , we construct a solution {(Yn,Zn,Vn,Mn)} as follows. By Theorem 2, on the interval [0,n] , Ytn=E(ξ∣Fn)+∫tn1[0,τ](s)f(s,Ysn,Zsn,Vsn)ds+∫tn1[0,τ](s)dRs−∫tnZsndWs−∫tn∫UVsn(e)πˆ(de,ds)−∫tndMsn, and for t≥n , we have by assumption (H7) that Ytn=ξt , Ztn=ηt , Vtn(e)=γt(e) , Mtn=Nt .

Step 1. We first show that (Yn,Zn,Vn,Mn) satisfies estimate (84). Applying Itô’s formula to epρs|Ysn|p over the interval [t∧τ,T∧τ] for 0≤t≤T≤n and combining with (42), (77), and (78) yield epρ(t∧τ)|Yt∧τn|p+αp∫t∧τT∧τepρs|Ysn|p−2|Zsn|2ds+αp∫t∧τT∧τepρs∫U|Ysn|p−2|Vsn(e)|2π(de,ds)+αp∫t∧τT∧τepρs|Ys−n|d[Mn]s≤epρ(T∧τ)|YT∧τn|p+p∫t∧τT∧τeρs[Ysn|Ysn|p−2f(s,Ysn,Zsn,Vsn)−ρ|Ysn|p]ds+p∫t∧τT∧τeρsYsn|Ysn|p−2dRs−p∫t∧τT∧τeρsYs−n|Ys−n|p−2dMsn−p∫t∧τT∧τeρsYsn|Ysn|p−2ZsndWs−p∫t∧τT∧τ∫UeρsYs−n|Ys−n|p−2Vsn(e)πˆ(de,ds). But, from the assumptions on f combined with Young’s inequality we get, for a small enough constant δ>0 0$]]>, y|y|p−2f(t,y,z,v)≤(μ+2pL2αp−pδ)|y|p+|y|p−1|f(t,0,0,0)|+(αpp−δ)|y|p−2|z|2+(αpp−δ)|y|p−2‖v‖Lλ2. Then, choosing δ such that μ+2pL2αp−pδ<ρ , we deduce from the above that epα(t∧τ)|Yt∧τn|p+pρ¯∫t∧τT∧τepαs|Ysn|pds+pδ∫t∧τT∧τepαs|Ysn|p−2|Zsn|2ds+αp∫t∧τT∧τepαs|Ysn|p−2|Vsn(e)|2π(de,ds)+αp∫t∧τT∧τepαs|Ysn|p−2d[Mn]s−p(αpp−δ)∫t∧τT∧τepαs|Ysn|p−2‖Vsn‖Lλ22ds≤epα(T∧τ)|YT∧τn|p+p∫t∧τT∧τepαs|Ysn|p−1|f(s,0,0,0)|ds+p∫t∧τT∧τeρs|Ysn|p−1d|R|s−p∫t∧τT∧τeρsYs−n|Ys−n|p−2dMsn−p∫t∧τT∧τeρsYsn|Ysn|p−2ZsndWs−p∫t∧τT∧τ∫UeρsYs−n|Ys−n|p−2Vsn(e)πˆ(de,ds). Note that all the local martingales in the last inequality are true martingales. Thus, taking the expectation in (87), we get in view of (81) that E[epρ(t∧τ)|Yt∧τn|p+pδ∫t∧τT∧τepρs|Ysn|pds+pδ∫t∧τT∧τepρs|Ysn|p−2|Zsn|2ds+pδ∫t∧τT∧τepρs|Ysn|p−2‖Vsn‖Lλ2ds+αp∫t∧τT∧τepρs|Ysn|p−2d[Mn]s]≤E[X], where X=epρ(T∧τ)|YT∧τn|p+p∫t∧τT∧τepρs|Ysn|p−1|f(s,0,0,0)|ds+p∫t∧τT∧τepρs|Ys−n|p−1d|R|s. Next, as in the proof of Lemma 4, including a sups∈[t,T] in (87) and applying the BDG inequality, we get in view of (88) that E[sups∈[t,T]epρ(s∧τ)|Ys∧τn|p]≤CpE[X]. But by Young’s inequality we have pE∫t∧τT∧τepρs|Ys|p−1|f(s,0,0,0)|ds≤pE(sups∈[t∧τ,T∧τ]e(p−1)ρs|Yt|p−1∫t∧τT∧τeρs|f(s,0,0,0)|ds)≤16E(sups∈[t∧τ,T∧τ]epρs|Yt|p)+dp′E(∫t∧τT∧τeρs|f(s,0,0,0)|ds)p, and pE∫t∧τT∧τepρs|Ys−|p−1d|R|s≤16E(sups∈[t∧τ,T∧τ]epρs|Yt|p)+dp″E(∫t∧τT∧τeρsd|R|s)p. Consequently, combining (89) with (90) and (91) and letting T→+∞ , we deduce that E[supt≥0epρ(t∧τ)|Yt∧τn|p]≤Cp″E[epρτ|ξ|p+(∫0τeρs|f(s,0,0,0)|ds)p+(∫0τeρsd|R|s)p]. Finally, going back to (88), we conclude in view of (90), (91), and (92) that estimate (84) holds for (Yn,Zn,Vn,Mn) .

Step 2. Let us show that (Yn,Zn,Vn,Mn) is a Cauchy sequence. For m>n n$]]>, define Y¯t=Ytm−Ytn,Z¯t=Ztm−Ztn,V¯t=Vtm−Vtn,M¯t=Mtm−Mtn. For n≤t≤m , we have Y¯t=∫t∧τm∧τf(s,Ysm,Zsm,Vsm)ds−∫t∧τm∧τZ¯sdWs−∫t∧τm∧τ∫UV¯s(e)πˆ(du,ds)−M¯m∧τ+M¯t∧τ. Consequently, again for n≤t≤m , epρ(t∧τ)|Y¯t∧τ|p+αp∫t∧τm∧τepρs|Y¯s|p−2|Z¯s|2ds+αp∫t∧τm∧τepρs∫U|Y¯s|p−2|V¯s(e)|2π(de,ds)+αp∫t∧τm∧τepρs|Y¯s−|d[M¯]s≤p∫t∧τm∧τeρs[Y¯s|Y¯s|p−2f(s,Ysm,Zsm,Vsm)−ρ|Y¯s|p]ds−p∫t∧τm∧τeρsY¯s|Y¯s|p−2Z¯sdWs−p∫t∧τm∧τeρsY¯s−|Y¯s−|p−2dM¯s−p∫t∧τm∧τ∫UeρsY¯s−|Y¯s−|p−2V¯s(e)πˆ(de,ds)≤p∫t∧τm∧τeρs[μ|Y¯s|p+L|Y¯s|p−1|Z¯s|+L|Y¯s|p−1‖V¯s(e)‖Lλ2−ρ|Y¯s|p]ds+p∫t∧τm∧τeρsY¯s|Y¯s|p−2f(s,ξs,ηs,γs)ds−p∫t∧τm∧τeρsY¯s|Y¯s|p−2Z¯sdWs−p∫t∧τm∧τeρsY¯s−|Y¯s−|p−2dM¯s−p∫t∧τm∧τ∫UeρsY¯s−|Y¯s−|p−2V¯s(e)πˆ(de,ds). We deduce by already used arguments that E[supn≤t≤mepρ(t∧τ)|Y¯t∧τ|p+∫n∧τm∧τepρs|Y¯s|pds+∫n∧τm∧τepρs|Y¯s|p−2|Z¯s|2ds+∫n∧τm∧τepρs∫U|Y¯s|p−2‖V¯s(e)‖Lλ22ds+∫n∧τm∧τepρs|Y¯s−|d[M¯]s]≤CE(∫n∧ττeρs|f(s,ξs,ηs,γs)|ds)p, and the last term tends to zero as n→∞ .

Next, for t≤n , Y¯t=Y¯n+∫t∧τn∧τ(f(s,Ysm,Zsm,Vsm)−f(s,Ysn,Zsn,Vsn))ds−∫t∧τn∧τZ¯sdWs−∫t∧τn∧τ∫UVˆs(e)πˆ(de,ds)−M¯n∧τ+M¯t∧τ. Arguing as in Proposition 2, we get Eepρ(t∧τ)|Y¯t∧τ|pE∫0τepρs|Y¯s|pds≤Eepρ(n∧τ)|Y¯n|p≤CE(∫n∧ττeρs|f(s,ξs,ηs,γs)|ds)p, and letting n→∞ , the convergence of the sequence Yn follows.

Next, it remains to show the convergence of the martingale part (Zn,Vn,Mn) . We follow the proof of Lemma 4. We apply Itô’s formula to e2ρs|Y¯s|2 for n≤t≤m : ∫t∧τm∧τe2ρs|Z¯s|2ds+∫t∧τm∧τ∫Ue2ρs|V¯s(e)|2π(de,ds)+∫t∧τm∧τe2ρsd[M¯]s=2∫t∧τm∧τe2ρs[Y¯s(f(s,Ysm,Zsm,Vsm)−f(s,ξs,ηs,γs))−ρ|Y¯s|2]ds+2∫t∧τm∧τe2ρsY¯sf(s,ξs,ηs,γs)ds−2∫0τne2ρsY¯sZ¯sdWs−2∫t∧τm∧τ∫Ue2ρsY¯s−V¯s(e)πˆ(de,ds)−2∫t∧τm∧τe2ρsY¯s−dM¯s. Mimicking the same argumentation used to obtain (34) (assumptions on f and BDG and Young inequalities) leads to E[(∫n∧τm∧τe2ρs|Z¯s|2ds)p2+(∫n∧τm∧τ∫Ue2ρs|V¯s(e)|2λ(de)ds)p2+(∫n∧τm∧τe2ρsd[M¯]s)p2]≤C(p,T)[Esupt≥nepρs|Y¯t|p+E(∫n∧ττeρs|f(s,ξs,ηs,γs)|ds)p]. Next, arguing similarly for the case t≤n , we get E[(∫0n∧τe2ρs|Z¯s|2ds)p2+(∫0n∧τ∫Ue2ρs|V¯s(e)|2λ(de)ds)p2+(∫0n∧τe2ρsd[M¯]s)p2]≤C(p,T)[Esupt≥nepρn∧τ|Y¯n∧τ|p]. Consequently, letting n→∞ , we deduce from the above that, in both cases, the sequence (Zn,Vn,Mn) is a Cauchy sequence for the norm E(∫0τe2ρs|Z¯s|2ds)p2+E(∫0τ∫Ue2ρs|V¯s(e)|2λ(de)ds)p2+E(∫0τe2ρsd[M¯]s)p2, and thus it converges to (Z,V,M) . Finally, the limit (Y,Z,V,M) is a solution of GBSDE (73) that satisfies estimates (84) and (85).  □

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