MSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania MSTA155 10.15559/20-MSTA155 Research Article Irregular barrier reflected BDSDEs with general jumps under stochastic Lipschitz and linear growth conditions https://orcid.org/0000-0002-0014-1458 MarzougueMohamedmohamed.marzougue@edu.uiz.ac.maa∗ SagnaYayasagnayaya88@gmail.comb Laboratory of Analysis and Applied Mathematics (LAMA), Faculty of sciences Agadir, Ibn Zohr University, 80000, Agadir, Morocco LERSTAD, UFR de Sciences Appliquées et de Technologie, Université Gaston Berger, BP 234, Saint-Louis, Senegal Corresponding author. 2020 106202072157190 632020 552020 2952020 © 2020 The Author(s). Published by VTeX2020 Open access article under the CC BY license.

In this paper, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.

 Reflected backward doubly stochastic differential equations irregular barrier Mertens decomposition stochastic Lipschitz condition stochastic linear growth condition 60H20 60H30 CNRST The corresponding author (Mohamed Marzougue) declares that this research was supported by National Center for Scientific and Technical Research (CNRST), Morocco. Introduction

Nonlinear backward stochastic differential equations (BSDEs in short) were first introduced by Pardoux and Peng [27] with the uniform Lipschitz condition under which they proved the celebrated existence and uniqueness result. Since then, the theory of BSDEs has been intensively developed in the last years. The great interest in this theory comes from its connections with many other fields of research, such as mathematical finance [12, 11], stochastic control and stochastic games [10] and partial differential equations [28]. After Pardoux and Peng introduced the theory of BSDEs, they considered [29] a new kind of BSDEs, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals with respect to two independent Brownian motions. They proved the existence and uniqueness of solutions to BDSDEs under uniform Lipschitz conditions on the coefficients.

In the setting of reflected BSDEs (resp. BDSDEs), an additional nondecreasing process is added in order to keep the solution above a certain lower-boundary process, called barrier (or obstacle), and to do this in a minimal fashion. The reflected BSDEs (RBSDEs in short) were introduced by El Karoui et al. [13], again under the uniform Lipschitz condition on the coefficients. The authors of [13] proved the existence and uniqueness results in the case of a Brownian filtration and a continuous barrier. The reflected BDSDEs (RBDSDEs in short) were introduced by Bahlali et al. [6] where the authors studied the case of RBDSDEs with continuous coefficients, and proved the existence and uniqueness of the solution.

To the best of our knowledge, the paper by Grigorova et al. [14] is the first one which studied RBSDEs in the case where the barrier is not necessarily right-continuous (just right upper semi-continuous). The authors of [14] studied the existence and uniqueness result under the Lipschitz assumption on the coefficients in a filtration that supports a Brownian motion and an independent Poisson random measure. Later, several authors have studied the RBSDEs following Grigorova et al. [14] (see e.g. [1–3, 17, 20, 23]). Recently, Berrhazi et al. [7] discussed the case of RBDSDE with a right upper semi-continuous barrier under Lipschitz coefficients.

Our aim in this paper is to extend the work on RBDSDEs with jumps (RBDSDEJs in short) to the case of an irregular barrier (which is assumed to be not necessarily right-continuous). The specificity of such equations lies in the fact that the two independent Brownian motions are coupled with an independent Poisson random measure. We’ll prove the existence and uniqueness of the solution to such equations under the so-called stochastic Lipschitz coefficients. The interest in this last condition is based on the fact that, unfortunately, in many applications, the usual Lipschitz conditions cannot be satisfied. For example, the pricing of the American claim is equivalent to solving the linear RBDSE −dVt=(rtVt+θtZt)dt−ZtdWt+dKt,VT=ξT;Vt≥ξt,(Vt−ξt)dKt=0a.s. where ξt is the amount received from the seller at time t, rt is the interest rate process and θt is the risk premium process. The additional process K is needed for this problem because there exists no replicating strategy for the option. We have to use a super-replicating strategy with a consumption process K. The minimality condition on K just states that we only invest money in the portfolio when Vt>ξt {\xi _{t}}$]]>. Here both rt and θt are not bounded in general. So, it is not possible to solve the RBSDE (1) by the result of El Karoui et al. [13]. Thus, in order to study more general RBSDEs (resp. RBDSDEs), one needs to relax the uniform Lipschitz conditions on the coefficients. To this direction, several attempts have been done. Among others, we refer to [4, 5, 9, 15, 21–24] for the case of BSDEs, and [16, 25, 26, 30] for BDSDEs.

In our paper, we use a generalization of the Doob–Meyer decomposition called the Mertens decomposition. This decomposition is used for strong optional supermartingales which are not necessarily right-continuous. We also use some tools from the optimal stopping theory, as well as a generalization of the Itô formula to the case of a strong optional supermartingale called the Gal’chouk–Lenglart formula due to Lenglart [19].

The paper is organized as follows. In Section 2, we give some notations, assumptions and main contributions needed in this paper. In Section 3, we prove the existence and uniqueness of the solution to RBDSDEJs with a stochastic Lipschitz coefficients (f,g) and an irregular barrier ξ, and we also give a comparison theorem for solutions. Section 4 is devoted to prove the existence of a minimal solution to RBDSDEJs under a stochastic growth coefficient f.

 Definitions and preliminary results

Let 0<T<+∞ be a non-random horizon time, Ω be a non-empty set, F be a σ-algebra of sets of Ω and P be a probability measure defined on F . The triple (Ω,F,P) defines a probability space which is assumed to be complete. We assume there are three mutually independent processes:

a d-dimensional Brownian motion (Wt)t≤T ,

a ℓ-dimensional Brownian motion (Bt)t≤T ,

a random Poisson measure μ on E×R+ with compensator ν(dt,de)=λ(de)dt , where the space E=Rℓ−{0} is equipped with its Borel field E such that {μ˜([0,t]×B)=(μ−ν)[0,t]×B} is a martingale for any B∈E satisfying λ(B)<∞ . λ is a σ-finite measure on E and satisfies ∫E(1∧|e|2)λ(de)<∞.

We consider the family (Ft)t≤T given by Ft=FtW∨Ft,TB∨Ftμ,0≤t≤T, where for any process (ηt)t≤T,Fs,tη=σ{ηr−ηs,s≤r≤t}∨N,Ftη=F0,tη . Here N denotes the class of P-null sets of F . Note that the family (Ft)t≤T does not constitute a classical filtration.

For an integer k≥1 , |.| and .,. stand for the Euclidian norm and the inner product in Rk , T[t,T] denotes the set of stopping times τ such that τ∈[t,T] and P denotes the σ-algebra of Ft -predictable sets of Ω×[0,T] .

For every Ft -measurable process (at)t≤T , we define an increasing process (At)t≤T by setting At=∫0tas2ds .

For every β>0 0$]]>, we consider the following sets (where E denotes the mathematical expectation with respect to the probability measure P):

S2(Rk) and Sβ2(Rk) are the spaces of Ft -adapted optional processes Ψ:Ω×[0,T]⟶Rk which satisfy, respectively, ΨS2(Rk)2=Eess supτ∈T[0,T]|Ψτ|2<+∞ and ΨSβ2(Rk)2=Eess supτ∈T[0,T]eβAτ|Ψτ|2<+∞.

M2(Rk×d) , Mβ2(Rk×d) and Mβ2,a(Rk) are the spaces of Ft -progressively measurable processes Ψ:Ω×[0,T]⟶Rk×d (resp. Rk ) which satisfy, respectively, ΨM2(Rk×d)2=E∫0T|Ψt|2dt<+∞,ΨMβ2(Rk×d)2=E∫0TeβAt|Ψt|2dt<+∞ and ΨMβ2,a(Rk)2=E∫0TeβAtat2|Ψt|2dt<+∞.

Lλ is the space of P⊗E -measurable mappings U:E⟶Rk such that Uλ2=∫E|U(e)|2λ(de)<+∞.

Lβ2(Rk) is the space of P⊗E -measurable processes U:Ω×[0,T]×E⟶Rk such that ULβ2(Rk)2=E∫0TeβAtUtλ2dt<+∞.

Notice that the space Aβ2(Rk)=Mβ2,a(Rk)×Mβ2(Rk×d)×Lβ2(Rk) endowed with the norm (Y,Z,U)Aβ2(Rk)2=YMβ2,a(Rk)2+ZMβ2(Rk×d)2+ULβ2(Rk)2 is a Banach space as is the space Bβ2(Rk)=(Mβ2,a(Rk)∩Sβ2(Rk))×Mβ2(Rk×d)×Lβ2(Rk) with the norm (Y,Z,U)Bβ2(Rk)2=YSβ2(Rk)2+YMβ2,a(Rk)2+ZMβ2(Rk×d)2+ULβ2(Rk)2.

For a làdlàg (limited from right and left) process (Yt)t≤T , we denote by:

Yt−=lims↗tYs the left-hand limit of Y at t∈[0,T] , (Y0−=Y0) , Y−:=(Yt−)t≤T and ΔYt:=Yt−Yt− the size of the left jump of Y at t.

Yt+=lims↘tYs the right-hand limit of Y at t∈[0,T] , (YT+=YT) , Y+:=(Yt+)t≤T and Δ+Yt:=Yt+−Yt the size of the right jump of Y at t.

Let f:Ω×[0,T]×Rk×Rk×d×Rk→Rk , g:Ω×[0,T]×Rk×Rk×d×Rk→Rk×ℓ , and (ξt)t≤T be an optional process which is assumed to be right upper semi-continuous and limited from left. The process (ξt)t≤T will be called irregular barrier. We are interested in the following RBDSDEJs associated with parameters (f,g,ξ) : Yτ=ξT+∫τTf(s,Θs)ds+∫τTg(s,Θs)dBs−∫τTZsdWs−∫τT∫EUs(e)μ˜(ds,de)+KT−Kτ+CT−−Cτ−τ∈T[0,T],Yτ≥ξτ∀τ∈T[0,T],K=Kc+Kd(continuous + purely discontinuous part) is anondecreasing right-continuous predictable process withK0=0such that∫0T1{Yt>ξt}dKtc=0a.s. and(Yτ−−ξτ−)ΔKτd=0a.s.∀τ∈T[0,T]p,Cis a nondecreasing right-continuous predictable purelydiscontinuous process withC0−=0such that(Yτ−ξτ)ΔCτ=0a.s.∀τ∈T[0,T]. {\xi _{t}}\}}}d{K_{t}^{c}}=0\hspace{2.5pt}\text{a.s. and}\hspace{2.5pt}({Y_{\tau -}}-{\xi _{\tau -}})\Delta {K_{\tau }^{d}}=0\hspace{2.5pt}\text{a.s.}\hspace{1em}\forall \tau \in {\mathcal{T}_{[0,T]}^{p}},\\ {} C\hspace{2.5pt}\text{is a nondecreasing right-continuous predictable purely}\\ {} \text{discontinuous process with}\hspace{2.5pt}{C_{0-}}=0\hspace{2.5pt}\text{such that}\hspace{2.5pt}\\ {} ({Y_{\tau }}-{\xi _{\tau }})\Delta {C_{\tau }}=0\hspace{2.5pt}\text{a.s.}\hspace{1em}\forall \tau \in {\mathcal{T}_{[0,T]}}.\end{array}\right.\]]]> Here Θs stands for the triple (Ys,Zs,Us) .

Let us consider the filtration (Gt)t≤T given by Gt=FtW∨FTB∨Ftμ,0≤t≤T which is assumed to be right-continuous and quasi-left-continuous, and make precise the notion of solution to RBDSDEJ (2).

Let ξ be an irregular barrier. A process (Y,Z,U,K,C) is called a solution to RBDSDEJ associated with parameters (f,g,ξ) , if it satisfies the system (2) and

(Y,Z,U)∈Bβ2(Rk) ,

(K,C)∈S2(Rk)×S2(Rk) .

We note that a process (Y,Z,U,K,C)∈Bβ2(Rk)×S2(Rk)×S2(Rk) satisfies the equation (2) if and only if Yt=ξT+∫tTf(s,Θs)ds+∫tTg(s,Θs)dBs−∫tTZsdWs−∫tT∫EUs(e)μ˜(ds,de)+KT−Kt+CT−−Ct−.

If (Y,Z,U,K,C) is a solution to RBDSDEJ (2), then ΔCt=Yt−Yt+ for all t≤T outside an evanescent set. It follows that Yt≥Yt+ for all t≤T , which implies that Y is necessarily right upper semi-continuous. Moreover, the process Yt+∫0tf(s,Θs)dst≤T is a strong supermartingale. Actually, by using Hölder’s inequality and the stochastic Lipschitz condition on f (below), we obtain, for each τ∈T[0,T] , EYτ+∫0τf(s,Θs)ds2≤2EYτ2+1βE∫0TeβAsf(s,Θs)as2ds≤2‖Y‖Sβ2(Rk)2+4β‖Y‖Mβ2,a(Rk)2+4β‖Z‖Mβ2(Rk×d)2+4β‖U‖Lβ2(Rk)2+4βf(.,0)aMβ2(Rk)2<+∞. Moreover, for all τ,ν∈T[0,T] with ν≤τ we have EYτ−Yν−∫ντf(s,Θs)ds|Gν=EKν−Kτ+Cν−−Cτ−|Gνa.s. Since K and C are nondecreasing processes, and Yt+∫0tf(s,Θs)dst≤T is a Gt -adapted process then the observation follows.

In our framework the filtration is quasi-left-continuous, martingales have only totally inaccessible jumps and Y has two type of left-jumps: totally inaccessible jumps which stem from stochastic integral with respect to μ˜ , and predictable jumps which come from the predictable jumps of the irregular barrier ξ. The latter are the source of the predictability of K. Moreover, the processes K and μ do not have jumps in common.

 (The particular case of a right-continuous barrier).

If the barrier ξ is right-continuous, we have Yt≥Yt+≥ξt+=ξt . Hence, if t is such that Yt=ξt , then Yt=Yt+=ξt . If t is such that Yt>ξt {\xi _{t}}$]]>, then by the minimality condition on C, Yt−Yt+=Ct−Ct−=0 . Thus, in both cases, Yt=Yt+ , so Y is right-continuous. Moreover, the right-continuity of Y combined with the fact that ΔCt=Yt−Yt+ give Ct=Ct− for all t≤T . As C is right-continuous, purely discontinuous and such that C0−=0 , we deduce C=0 . Thus, we recover the usual formulation of RBDSDEJs with a right-continuous barrier.

Let (Y,Z,U)∈Bβ2(Rk) with Y being a làdlàg process, and let a coefficient g(.)∈Mβ2(Rk×ℓ) . Then ∫0teβAs⟨Ys−,ZsdWs⟩+⟨Ys−,g(s)dBs⟩+∫E⟨Ys−,Us(e)μ˜(ds,de)⟩t≤T is a martingale.

Using the left-continuity of trajectories of the process Ys− , we have |Ys−(ω)|2≤supt∈[0,T]∩Q|Yt−(ω)|2∀(s,ω)∈[0,T]×Ω. On the other hand, we have |Yt−|2≤ess supτ∈T[0,T]|Yτ|2 which implies supt∈[0,T]∩Q|Yt−|2≤ess supτ∈T[0,T]|Yτ|2. Then for all τ≤T ∫0τe2βAs|Ys−|2|Zs|2ds≤∫0τe2βAssupt∈[0,T]∩Q|Yt−|2|Zs|2ds≤∫0τe2βAsess supτ∈T[0,T]|Yτ|2|Zs|2ds. Further, we have ∫0τe2βAsess supτ∈T[0,T]|Yτ|2|Zs|2ds≤ess supτ∈T[0,T]eβAτ|Yτ|2∫0τeβAs|Zs|2ds. Hence E∫0τe2βAs|Ys−|2|Zs|2ds≤Eess supτ∈T[0,T]eβAτ|Yτ|2∫0TeβAs|Zs|2ds≤12‖Y‖Sβ2(Rk)2+‖Z‖Mβ2(Rk×d)2. Since (Y,Z)∈Sβ2(Rk)×Mβ2(Rk×d) , we get the finite expectation. Since the process ∫0teβAs⟨Ys,ZsdWs⟩t≤T is adapted, it is a martingale.

By the same arguments, ∫0t∫EeβAs⟨Ys−,Us(e)μ˜(ds,de)⟩t≤Tand∫0teβAs⟨Ys−,g(s)dBs⟩t≤T are martingales since E∫0τ∫Ee2βAs|Ys−|2|Us(e)|2λ(de)ds≤Eess supτ∈T[0,T]eβAτ|Yτ|2∫0TeβAs‖Us‖λ2ds≤12‖Y‖Sβ2(Rk)2+‖U‖Lβ2(Rk)2 and E∫0τe2βAs|Ys−|2|g(s)|2ds≤Eess supτ∈T[0,T]eβAτ|Yτ|2∫0TeβAs|g(s)|2ds≤12‖Y‖Sβ2(Rk)2+‖g‖Mβ2(Rk×ℓ)2.  □

Let us recall some results from the general theory of optional processes, which will be useful in the sequel.

(Mertens decomposition).

Let Y˜ be a strong optional supermartingale of class (D). There exists a unique uniformly integrable martingale (càdlàg) N, a unique nondecreasing right-continuous predictable process K with K0=0 and E|KT|2<+∞ , and a unique nondecreasing right-continuous adapted purely discontinuous process C with C0−=0 and E|CT|2<+∞ , such that Y˜t=Nt−Kt−Ct−∀t≤Ta.s.

(Dellacherie–Meyer).

Let K be a nondecreasing predictable process. Let U be the potential of the process K, i.e. U:=E[KT|Gt]−Kt for all t≤T . We assume that there exists a positive GT -measurable random variable X such that |Uν|≤E[X|Gν] a.s. for all ν∈T[0,T] . Then E|KT|2≤cE|X|2 , where c is a positive constant.

The proof is established in chapter VI, Theorem 99, [8] for the case of a nondecreasing process which is not necessarily right-continuous nor left-continuous.

Let Y be a strong optional supermartingale of class (D) such that, for all ν∈T[0,T] , |Yν|≤E[X|Gν] a.s., where X is a nonnegative GT -measurable random variable. Let K˜ be the Mertens process associated with Y. Then there exists a positive constant c such that E|K˜T|2≤cE|X|2 .

The proof is established in [23]. (Gal’chouk–Lenglart formula).

Let n∈N . Let Y be an n-dimensional optional semimartingale with the decomposition Yk=Y0k+Mk+Rk+Ok , for all k=1,…,n , where Mk is a (càdlàg) local martingale, Rk is a right-continuous process of finite variation such that R0k=0 and Ok is a left-continuous process of finite variation which is purely discontinuous and such that O0k=0 . Let F be a twice continuously differentiable function on Rn . Then, almost surely, for all t≥0 , F(Yt)=F(Y0)+ ∑k=1n∫0tDkF(Ys−)d(Mk+Rk)s+ ∑k=1n∫0tDkF(Ys)dOs+k+12 ∑k,l=1n∫0tDkDlF(Ys−)d[Mk,c,Ml,c]s+∑0<s≤tF(Ys)−F(Ys−)− ∑k=1nDkF(Ys−)ΔYsk+∑0≤s<tF(Ys+)−F(Ys)− ∑k=1nDkF(Ys)Δ+Ysk, where Dk denotes the differentiation operator with respect to the k-th coordinate, and Mk,c denotes the continuous part of Mk .

Let Y be an optional semimartingale with the decomposition Y=Y0+M+R+O where M, R and O are as in Theorem 2.9. Then, almost surely, for all t≥0 , eβAt|Yt|2+β∫tTeβAsas2|Ys|2ds=eβAT|YT|2+2∫tTeβAsYs−d(M+R)s+∫tTeβAsYsdOs++∫tTeβAsd[Mc,Mc]s−∑t<s≤TeβAsΔYs2−∑t≤s<TeβAsΔ+Ys2.

To prove the corollary, it suffices to apply the change of variables formula from Theorem 2.9 with F(X,Y)=XY2 for Xt=eβAt .  □

Let Y∈Sβ2(Rk),ϑ∈Mβ2(Rk),ζ∈Mβ2(Rk×ℓ),π∈Mβ2(Rk×d) and ϕ∈Lβ2(Rk) be such that Yt=Y0−∫0tϑsds−∫0tζsdBs+∫0tπsdWs+∫0t∫Eϕs(e)μ˜(ds,de)−Kt−Ct−, where E|KT|2+E|CT|2<+∞ . Then Y is an optional semimartingale with the decomposition Y=Y0+M+R+O where Mt=−∫0tζsdBs+∫0tπsdWs+∫0t∫Eϕs(e)μ˜(ds,de) , Rt=−∫0tϑsds−Kt and Ot=−Ct− , and we have, for any β>0 0$]]> and t≤T , eβAt|Yt|2+β∫tTeβAsas2|Ys|2ds+∫tTeβAs|πs|2ds=eβAT|YT|2+2∫tTeβAs⟨Ys−,ϑs⟩ds+2∫tTeβAs⟨Ys−,ζsdBs⟩−2∫tTeβAs⟨Ys−,πsdWs⟩−2∫tT∫EeβAs⟨Ys−,ϕs(e)μ˜(de,ds)⟩+∫tTeβAs|ζs|2ds+2∫tTeβAs⟨Ys−,dKs⟩+2∫tTeβAs⟨Ys,dCs⟩−∑t<s≤TeβAsΔYs2−∑t≤s<TeβAsΔ+Ys2.

Reflected BDSDEJs with stochastic Lipschitz coefficients Assumptions

We assume that the parameters (f,g,ξ) satisfy the following assumptions (A1), for some β>0 0$]]> (where we define for all t≤T,h(t,0)=h(t,0,0,0),forh∈f,g to ease the reading).

f and g are jointly measurable, and there exists a constant α∈]0,1[ and four non-negative, FtW -measurable processes (γt)t≤T , (κt)t≤T , (σt)t≤T and (ϱt)t≤T such that for all (y,y′)∈(Rk)2 , (z,z′)∈(Rk×d)2 and (u,u′)∈(Lλ)2 , |f(t,y,z,u)−f(t,y′,z′,u′)|≤γt|y−y′|+κt|z−z′|+σtu−u′λ,|g(t,y,z,u)−g(t,y′,z′,u′)|2≤ϱt|y−y′|2+α|z−z′|2+u−u′λ2.

For all 0≤t≤T,at2=γt+κt2+σt2+ϱt>0 0$]]>.

For any (t,y,z,u)∈[0,T]×Rk×Rk×d×Lλ,f(t,y,z,u) and g(t,y,z,u) are Ft -measurable with f(.,0)a∈Mβ2(Rk) and g(.,0)∈Mβ2(Rk×ℓ) .

The irregular barrier (ξt)t≤T is in S2β2(Rk) .

Existence and uniqueness of solution

Before proving the existence and uniqueness, let us establish the corresponding result in the case where the coefficients f and g do not depend on the variables Y, Z and U. So we consider the RBDSDEJ, ∀τ∈T[0,T] , Yτ=ξT+∫τTf(s)ds+∫τTg(s)dBs−∫τTZsdWs−∫τT∫EUs(e)μ˜(ds,de)+KT−Kτ+CT−−Cτ−,Yτ≥ξτ,∫0T1{Yt>ξt}dKtc=0,(Yτ−−ξτ−)ΔKτd=0and(Yτ−ξτ)ΔCτ=0a.s. {\xi _{t}}\}}}d{K_{t}^{c}}=0,\hspace{1em}({Y_{\tau -}}-{\xi _{\tau -}})\Delta {K_{\tau }^{d}}=0\hspace{1em}\text{and}\hspace{1em}({Y_{\tau }}-{\xi _{\tau }})\Delta {C_{\tau }}=0\hspace{2.5pt}\text{a.s.}\end{array}\right.\]]]> where K=Kc+Kd (continuous + purely discontinuous part) is a nondecreasing right-continuous predictable process with K0=0 and C is a nondecreasing right-continuous predictable purely discontinuous process with C0−=0 . Moreover, the irregular barrier ξ satisfies (A1.4) and the coefficients (f,g) satisfy the following condition:

For any t≤T , f(t) and g(t) are Ft -measurable with f(.)a∈Mβ2(Rk) and g(.)∈Mβ2(Rk×ℓ) .

Let us prove an a priori estimate of the solution in the following lemma.

Let (Y1,Z1,U1,K1,C1) and (Y2,Z2,U2,K2,C2) be two solutions to RBDSDEJs with parameters (f1(.),g1(.),ξ1) and (f2(.),g2(.),ξ2) , respectively. We denote ℜ‾:=ℜ1−ℜ2 for ℜ∈{Y,Z,U,K,C,f,g,ξ} . Then there exists a constant κ(β) depending on β such that for all β>1 1$]]> ‖Y‾‖Bβ2(Rk)2+‖Z‾‖Mβ2(Rk×d)2+‖U‾‖Lβ2(Rk)2≤κ(β)‖ξ‾‖S2β2(Rk)2+f‾aMβ2(Rk)2+g‾Mβ2(Rk×ℓ)2.

Let τ∈T[0,T] . It is obvious that the process Y‾ is an optional semimartingale with the decomposition Y‾τ=Y‾0+Mτ+Rτ+Oτ where Mτ=−∫0τg‾(s)dBs+∫0τZ‾sdWs+∫0τ∫EU‾s(e)μ˜(ds,de) , Rτ=−∫0τf‾(s)ds−K‾τ and Oτ=−C‾τ− . Then, from Lemma 2.11, we have eβAt|Y‾t|2+β∫tTeβAsas2|Y‾s|2ds+∫tTeβAs|Z‾s|2ds=eβAT|ξ‾T|2+2∫tTeβAs⟨Y‾s−,f‾(s)⟩ds+2∫tTeβAs⟨Y‾s−,dK‾s⟩−2∫tTeβAs⟨Y‾s−,Z‾sdWs⟩−2∫tT∫EeβAs⟨Y‾s−,U‾s(e)μ˜(ds,de)⟩+2∫tTeβAs⟨Y‾s−,g‾(s)dBs⟩+∫tTeβAs|g‾(s)|2ds+2∫tTeβAs⟨Y‾s,dC‾s⟩−∑t<s≤TeβAs(ΔY‾s)2−∑t≤s<TeβAs(Δ+Y‾s)2. From Remark 2.3, the processes K‾ and μ do not have jumps in common, but K‾ jumps at predictable stopping times and μ jumps only at totally inaccessible stopping times. Then we can note that ∑t<s≤TeβAs(ΔY‾s)2=∫tT∫EeβAs|U‾s(e)|2μ(ds,de)+∑t<s≤TeβAs(ΔK‾s)2. Hence ∫tTeβAs‖U‾s‖λ2ds−∑t<s≤TeβAs(ΔY‾s)2=∫tTeβAs‖U‾s‖λ2ds−∫tT∫EeβAs|U‾s(e)|2μ(ds,de)−∑t<s≤TeβAs(ΔK‾s)2≤−∫tT∫EeβAs|U‾s(e)|2μ˜(ds,de). On the other hand, by using the Skorokhod and minimality conditions on K‾ and C‾ we can show that ⟨Y‾s−,dK‾s⟩≤0 and ⟨Y‾s,dC‾s⟩≤0 . Indeed, for all s≤T ⟨Y‾s−,dK‾s⟩=⟨Ys−1−ξs−,dKs1,c+ΔKs1,d⟩−⟨Ys−2−ξs−,dKs1,c+ΔKs1,d⟩−⟨Ys−1−ξs−,dKs2,c+ΔKs2,d⟩+⟨Ys−2−ξs−,dKs2,c+ΔKs2,d⟩=−⟨Ys−2−ξs−,dKs1,c+ΔKs1,d⟩−⟨Ys−1−ξs−,dKs2,c+ΔKs2,d⟩≤0,sinceYi≥ξfori=1,2. Furthermore we have ⟨Y‾s,dC‾s⟩=⟨Y‾s,ΔC‾s⟩ , and by the same arguments, we have, for all s≤T , ⟨Y‾s,ΔC‾s⟩=⟨Ys1−ξs,ΔCs1⟩−⟨Ys2−ξs,ΔCs1⟩−⟨Ys1−ξs,ΔCs2⟩−⟨ξs−Ys2,ΔCs2⟩=0−⟨Ys2−ξs,ΔCs1⟩−⟨Ys1−ξs,ΔCs2⟩−0≤0,sinceYi≥ξfori=1,2. Moreover, by using the fact that 2⟨Y‾s,f‾(s)⟩≤(β−1)as2|Y‾s|2+1β−1|f‾(s)|2as2∀β>1, 1,\]]]> the inequality (4) becomes eβAt|Y‾t|2+∫tTeβAsas2|Y‾s|2ds+∫tTeβAs|Z‾s|2ds+∫tTeβAs‖U‾s‖λ2ds≤ess supτ∈T[0,T]e2βAτ|ξ‾τ|2+1β−1∫tTeβAsf‾(s)as2ds−2∫tTeβAs⟨Y‾s−,Z‾sdWs⟩−2∫tT∫EeβAs⟨Y‾s−,U‾s(e)μ˜(ds,de)⟩+2∫tTeβAs⟨Y‾s−,g‾(s)dBs⟩+∫tTeβAs|g‾(s)|2ds. Taking the expectation on the both sides of the inequality (5) and using Proposition 2.5, we get, for all β>1 1$]]>, ‖Y‾‖Mβ2,a(Rk)2+‖Z‾‖Mβ2(Rk×d)2+‖U‾‖Lβ2(Rk)2≤‖ξ‾‖S2β2(Rk)2+1β−1f‾aMβ2(Rk)2+g‾Mβ2(Rk×ℓ)2. On the other hand, by taking the essential supremum over τ∈T[0,T] and then the expectation on both sides of inequality (5) we obtain Eess supτ∈T[0,T]eβAτ|Y‾τ|2≤Eess supτ∈T[0,T]e2βAτ|ξ‾τ|2+1β−1E∫0TeβAsf‾(s)as2ds+2Eess supτ∈T[0,T]∫0τeβAs⟨Y‾s−,Z‾sdWs⟩+2Eess supτ∈T[0,T]∫0τ∫EeβAs⟨Y‾s−,U‾s(e)μ˜(ds,de)⟩+2Eess supτ∈T[0,T]∫0τeβAs⟨Y‾s−,g‾(s)dBs⟩+∫0TeβAs|g‾(s)|2ds. From the Burkhölder–Davis–Gundy inequality, there exists a universal constant c such that 2Eess supτ∈T[0,T]∫0τeβAs⟨Y‾s−,Z‾sdWs⟩≤2cE∫0Te2βAs|Y‾s−|2|Z‾s|2ds≤14‖Y‾‖Sβ2(Rk)2+4c2‖Z‾‖Mβ2(Rk×d)2, 2Eess supτ∈T[0,T]∫0τ∫EeβAs⟨Y‾s−,U‾s(e)μ˜(ds,de)⟩≤2cE∫0Te2βAs|Y‾s−|2‖U‾s‖λ2ds≤14‖Y‾‖Sβ2(Rk)2+4c2‖U‾‖Lβ2(Rk)2 and 2Eess supτ∈T[0,T]∫0τeβAs⟨Y‾s−,g‾(s)dBs⟩≤2cE∫0Te2βAs|Y‾s−|2|g‾(s)|2ds≤14‖Y‾‖Sβ2(Rk)2+4c2‖g‾‖Mβ2(Rk×ℓ)2. Consequently, ‖Y‾‖Sβ2(Rk)2≤4‖ξ‾‖S2β2(Rk)2+1β−1f‾aMβ2(Rk)2+(4c2+1)g‾Mβ2(Rk×ℓ)2+4c2‖Z‾‖Mβ2(Rk×d)2+4c2‖U‾‖Lβ2(Rk)2. The desired result is obtained by combining the estimates (6) and (7) for β>1 1$]]>.  □

In the following, we state the existence and uniqueness result for the solution to RBDSDEJ (3).

Under the assumptions (A1.4) and (A1.5), the RBDSDEJ (3) admits a unique solution (Y,Z,U,K,C)∈Bβ2(Rk)×S2(Rk)×S2(Rk) for all β>1 1$]]>, and for each ν∈T[0,T] we have Yν=ess supτ∈T[ν,T]Eξτ+∫ντf(t)dt+∫ντg(t)dBt|Gνa.s.

Let ν∈T[0,T] . We define the value function Y‾‾(ν) by Y‾‾(ν)=ess supτ∈T[ν,T]Eξτ+∫ντf(t)dt+∫ντg(t)dBt|Gν, and Y˜(ν) by Y˜(ν)=Y‾‾(ν)+∫0νf(t)dt+∫0νg(t)dBt=ess supτ∈T[ν,T]Eξτ+∫0τf(t)dt+∫0τg(t)dBt|Gν. The process ξt+∫0tf(s)ds+∫0tg(s)dBst≤T is progressively measurable. Therefore, the family (Y˜(ν))ν∈T[0,T] is a supermartingale family. This observation with the Remark b. page 435 in [8] ensures the existence of a strong optional supermartingale Y˜ such that Y˜ν=Y˜(ν) for all ν∈T[0,T] . Thus, we have Y‾‾(ν)=Y˜ν−∫0νf(t)dt−∫0νg(t)dBt . On the other hand, almost all trajectories of the strong optional supermartingale are làdlàg, then the làdlàg optional process (Y‾‾t)t≤T:=Y˜t−∫0tf(s)ds−∫0tg(s)dBst≤T aggregates the family (Y‾‾(ν))ν∈T[0,T] .

Now, it remains to show that the candidate Y‾‾∈Sβ2(Rk) . Using the Jensen’s, Young’s and Hölder’s inequalities respectively, we obtain eβ2Aν|Y‾‾ν|=|ess supτ∈T[ν,T]E[eβ2Aνξτ+eβ2Aν∫ντf(t)dt+eβ2Aν∫ντg(t)dBt|Gν]|≤E[{|ess supτ∈T[ν,T]eβ2Aνξτ+eβ2Aν∫νTf(t)dt+ess supτ∈T[ν,T]eβ2Aν∫ντg(t)dBt|2}12|Gν]≤3E[{ess supτ∈T[ν,T]eβAτ|ξτ|2+eβAν|∫νTf(t)dt|2+ess supτ∈T[ν,T]eβAν|∫ντg(t)dBt|2}12|Gν]≤3E[{ess supτ∈T[ν,T]e2βAτ|ξτ|2+eβAν(∫νTe−βAtat2dt)×(∫νTeβAt|f(t)at|2dt)+c∫0TeβAt|g(t)|2dt}12|Gν]≤3E[{ess supτ∈T[0,T]e2βAτ|ξτ|2+1β∫0TeβAt|f(t)at|2dt+c∫0TeβAt|g(t)|2dt}12|Gν]. Taking the essential supremum over ν∈T[0,T] on the above sides and using the Doob’s martingale inequality, we conclude that Eess supν∈T[0,T]eβAν|Y‾‾ν|2≤κ′(β)Eess supτ∈T[ν,T]e2βAτ|ξτ|2+∫0TeβAtf(t)at2dt+∫0TeβAt|g(t)|2dt where κ′(β) is a positive constant depending on β. It follows that Y‾‾∈Sβ2(Rk) .

Note that the strong optional supermartingale Y˜ is of class (D) (i.e. the set of all random variables Y˜ν , for each finite stopping time ν, is uniformly integrable). Then by the Mertens decomposition (see Theorem 2.6), there exists a uniformly integrable martingale (càdlàg) N, a nondecreasing right-continuous predictable process K (with K0=0 ) such that E|KT|2<+∞ and a nondecreasing right-continuous adapted purely discontinuous process C (with C0−=0 ) such that E|CT|2<+∞ , with the following equality: Y˜τ=Nτ−Kτ−Cτ−∀τ∈T[0,T]. By an extension of Itô’s martingale representation Theorem, there exists a unique pair of predictable processes (Z,U)∈M2(Rk×d)×L2(Rk) such that Nτ=N0+∫0τZsdWs+∫0τ∫EUs(e)μ˜(ds,de). Hence for each τ∈T[0,T] Y‾‾τ=−∫0τf(s)ds−∫0τg(s)dBs+N0+∫0τZsdWs−∫0τ∫EUs(e)μ˜(ds,de)−Kτ−Cτ− with Y‾‾T=Y‾‾(T)=ξT and Y‾‾τ=Y‾‾(τ)≥ξτ a.s for all τ∈T[0,T] . Next, let us focus on the Skorokhod and minimality conditions. Since Δ+Y‾‾τ=1{Y‾‾τ=ξτ}Δ+Y‾‾τ a.s.(see Remark A.4 in [14]), from (8) we have ΔCτ=−Δ+Y‾‾τ a.s., then ΔCτ=1{Y‾‾τ=ξτ}ΔCτ a.s. It follows that the minimality condition on C is satisfied. Further, due to a result from the optimal stopping theory (see Proposition B.11 in [18]), for each predictable stopping time τ, we have ∫0T1{Y‾‾t>ξt}dKtc=0 {\xi _{t}}\}}}d{K_{t}^{c}}=0$]]> a.s. and ΔKτd=1{Y‾‾τ−=ξτ−}ΔKτd a.s. Then the process K satisfies the Skorokhod condition. Thus, we found a process (Y‾‾,Z,U,K,C) which satisfies the RBDSDEJ (3).

Now, it remains to show that (Y‾‾,Z,U,K,C)∈Bβ2(Rk)×S2(Rk)×S2(Rk) . Indeed, let K˜t:=Kt+Ct− be the Mertens process associated with Y˜ . By the definition of Y˜ν , we see that |Y˜ν|=ess supτ∈T[ν,T]Eξτ+∫0τf(t)dt+∫0τg(t)dBt|Gν≤Eess supτ∈T[ν,T]|ξτ|+∫0T|f(t)|dt+ess supτ∈T[ν,T]∫0τg(t)dBt|Gν. From Corollary 2.8, there exists a positive constant c such that E|K˜T|2≤cEess supτ∈T[ν,T]|ξτ|+∫0T|f(t)|dt+ess supτ∈T[ν,T]∫0τg(t)dBt2≤c(β)‖ξ‖S2β2(Rk)2+faMβ2(Rk)2+gMβ2(Rk×ℓ)2 where c(β) is a positive constant depending on β. K˜ is nondecreasing, and it implies that Eess supν∈T[0,T]|K˜τ|2≤E|K˜T|2<+∞. It follows that K˜∈S2(Rk) , then (K,C)∈S2(Rk)×S2(Rk) . On the other hand, from Lemma 2.11 we have eβAt|Y‾‾t|2+β∫tTeβAsas2|Y‾‾s|2ds+∫tTeβAs|Zs|2ds=eβAT|ξT|2+2∫tTeβAs⟨Y‾‾s−,f(s)⟩ds+2∫tTeβAs⟨Y‾‾s−,dKs⟩−2∫tTeβAs⟨Y‾‾s−,ZsdWs⟩−2∫tT∫EeβAs⟨Y‾‾s−,Us(e)μ˜(ds,de)⟩+2∫tTeβAs⟨Y‾‾s−,g(s)dBs⟩+∫tTeβAs|g(s)|2ds+2∫tTeβAs⟨Y‾‾s,dCs⟩−∑t<s≤TeβAs(ΔY‾‾s)2−∑t≤s<TeβAs(Δ+Y‾‾s)2. From Remark 2.3, the processes K and μ do not have jumps in common, but K jumps at predictable stopping times and μ jumps only at totally inaccessible stopping times, then we can write ∑t<s≤TeβAs(ΔY‾‾s)2=∫tT∫EeβAs|Us(e)|2μ(ds,de)+∑t<s≤TeβAs(ΔKs)2. Hence ∫tTeβAs‖Us‖λ2ds−∑t<s≤TeβAs(ΔY‾‾s)2=∫tTeβAs‖Us‖λ2ds−∫tT∫EeβAs|Us(e)|2μ(ds,de)−∑t<s≤TeβAs(ΔKs)2≤−∫tT∫EeβAs|Us(e)|2μ˜(ds,de). Consequently, ∫tTeβAsas2|Y‾‾s|2ds+∫tTeβAs|Zs|2ds+∫tTeβAs‖Us‖λ2ds≤eβAT|ξT|2+1β−1∫tTeβAsf(s)as2ds−2∫tTeβAs⟨Y‾‾s−,ZsdWs⟩−2∫tT∫EeβAs⟨Y‾‾s−,Us(e)μ˜(ds,de)⟩+2∫tTeβAs⟨Y‾‾s−,g(s)dBs⟩+∫tTeβAs|g(s)|2ds+2ess supτ∈T[0,T]e2βAτ|ξτ|2+KT2+CT2. Here we have used also the Skorokhod and minimality conditions on K and C. Next, by taking the expectation on both sides of above inequality, we get ‖Y‾‾‖Mβ2,a(Rk)2+‖Z‖Mβ2(Rk×d)2+‖U‖Lβ2(Rk)2≤3‖ξ‖S2β2(Rk)2+1β−1faMβ2(Rk)2+gMβ2(Rk×ℓ)2+E|KT|2+E|CT|2. Then (Y‾‾,Z,U)∈Mβ2,a(Rk)×Mβ2(Rk×d)×Lβ2(Rk) .

Finally, it is remarkabe that the uniqueness of the solution comes from the uniqueness of the Mertens decomposition and the Itô’s martingale representation Theorem, and if Y‾‾ and Y are two first-components of the solution, then by Lemma 3.1 we have immediately Y‾‾=Y .  □

Assume that the assumptions (A1.1)–(A1.4) are true. Then, if (y,z,u)∈Bβ2(Rk) for β>1 1$]]>, there exists a unique process (Y,Z,U,K,C)∈Bβ2(Rk)×S2(Rk)×S2(Rk) being a solution to the following RBDSDEJ, for all τ∈T[0,T] , Yτ=ξT+∫τTf(s,ys,zs,us)ds+∫τTg(s,ys,zs,us)dBs−∫τTZsdWs−∫τT∫EUs(e)μ˜(ds,de)+KT−Kτ+CT−−Cτ−,Yτ≥ξτ,∫0T1{Yt>ξt}dKtc=0,(Yτ−−ξτ−)ΔKτd=0and(Yτ−ξτ)ΔCτ=0a.s. {\xi _{t}}\}}}d{K_{t}^{c}}=0,\hspace{1em}({Y_{\tau -}}-{\xi _{\tau -}})\Delta {K_{\tau }^{d}}=0\hspace{1em}\textit{and}\hspace{1em}({Y_{\tau }}-{\xi _{\tau }})\Delta {C_{\tau }}=0\hspace{0.2778em}a.s.\end{array}\right.\]]]>

Given (y,z,u)∈Bβ2(Rk) , we define fˆ(t)=f(t,yt,zt,ut) and gˆ(t)=g(t,yt,zt,ut) . Let us show that fˆ and gˆ satisfy (A1.5). From the assumptions (A1.1) and (A1.2), we have |fˆ(s)|2≤4as4|ys|2+as2|zs|2+as2‖us‖λ2+|f(s,0)|2 and |gˆ(s)|2≤2as2|ys|2+α(|zs|2+‖us‖λ2)+|g(s,0)|2. Thus gathering these inequalities, we deduce that E(∫0TeβAsfˆ(s)as2ds+∫0TeβAs|gˆ(s)|2ds)≤E6∫0TeβAsas2|ys|2ds+(4+2α)∫0TeβAs(|zs|2+‖us‖λ2)ds+E4∫0TeβAsf(s,0)as2ds+2∫0TeβAs|g(s,0)|2ds. This implies that fˆ and gˆ satisfy (A1.5) since (y,z,u)∈Bβ2(Rk) and in view of the assumption (A1.3). Hence the result follows from Proposition 3.2.  □

We are now in position to study the solvability of our RBDSDEJ (2) associated with parameters (f(.,Θ),g(.,Θ),ξ) .

Under the assumptions (A1.1)–(A1.4), there exists β0>0 0$]]> such that for all β≥β0 the RBDSDEJ (2) admits a unique solution (Y,Z,U,K,C)∈Bβ2(Rk)×S2(Rk)×S2(Rk) .

(i) Existence. Our strategy in the proof of existence is to use the Picard approximate sequence. To this end, we consider the sequence (Θn)n≥0:=(Yn,Zn,Un)n≥0∈Bβ2(Rk) defined recursively by Y0=Z0=U0=0 and for any n≥1 , τ∈T[0,T] , Yτn+1=ξT+∫τTfs,Θsnds+∫τTgs,ΘsndBs−∫τTZsn+1dWs−∫τT∫EUsn+1(e)μ˜(ds,de)+KTn+1−Kτn+1+CT−n+1−Cτ−n+1,Yτn+1≥ξτa.s.,∫0T1{Ytn+1>ξt}dKtc,n+1=0a.s.and(Yτ−n+1−ξτ−)ΔKτd,n+1=0a.s.,(Yτn+1−ξτ)ΔCτn+1=0a.s. {\xi _{t}}\}}}d{K_{t}^{c,n+1}}=0\hspace{2.5pt}\text{a.s.}\hspace{1em}\text{and}\hspace{1em}({Y_{\tau -}^{n+1}}-{\xi _{\tau -}})\Delta {K_{\tau }^{d,n+1}}=0\hspace{2.5pt}\text{a.s.,}\\ {} ({Y_{\tau }^{n+1}}-{\xi _{\tau }})\Delta {C_{\tau }^{n+1}}=0\hspace{2.5pt}\text{a.s.}\end{array}\right.\]]]> Since for n≥0,(Yn,Zn,Un)∈Bβ2(Rk) , by virtue of Proposition 3.3, RBDSDEJ (9) has a unique solution (Yn+1,Zn+1,Un+1,Kn+1,Cn+1)∈Bβ2(Rk)×S2(Rk)×S2(Rk) .

In the sequel, we shall show that (Yn,Zn,Un)n≥0 is a Cauchy sequence in the Banach space Bβ2(Rk) . We define ℜ‾n+1=ℜn+1−ℜn for ℜ∈{Y,Z,U,K,C} , and ∀h∈f,g,h‾Θn(t)=h(t,Θtn)−h(t,Θtn−1),t≤T. We derive that for any n≥1 the process (Y‾n+1,Z‾n+1,U‾n+1,K‾n+1,C‾n+1) satisfies the following equation Y‾tn+1=∫tTf‾Θn(s)ds+∫tTg‾Θn(s)dBs−∫tTZ‾sn+1dWs−∫tT∫EU‾sn+1(e)μ˜(ds,de)+K‾Tn+1−K‾tn+1+C‾T−n+1−C‾t−n+1. Applying the Lemma 2.11, we have eβAt|Y‾tn+1|2+β∫tTeβAsas2|Y‾sn+1|2ds+∫tTeβAs|Z‾sn+1|2ds=2∫tTeβAs⟨Y‾s−n+1,f‾Θn(s)⟩ds+2∫tTeβAs⟨Y‾s−n+1,dK‾sn+1⟩−2∫tTeβAs⟨Y‾s−n+1,Z‾sn+1dWs⟩−2∫tT∫EeβAs⟨Y‾s−n+1,U‾sn+1(e)μ˜(ds,de)⟩+2∫tTeβAs⟨Y‾s−n+1,g‾Θn(s)dBs⟩+∫tTeβAs|g‾Θn(s)|2ds+2∫tTeβAs⟨Y‾sn+1,dC‾sn+1⟩−∑t<s≤TeβAs(ΔY‾sn+1)2−∑t≤s<TeβAs(Δ+Y‾sn+1)2. From Remark 2.3, the processes K‾n+1 and μ do not have jumps in common, but K‾n+1 jumps at predictable stopping times and μ jumps only at totally inaccessible stopping times, then ∫tTeβAs‖U‾sn+1‖λ2ds−∑t<s≤TeβAs(ΔY‾sn+1)2≤−∫tT∫EeβAs|U‾sn+1(e)|2μ˜(ds,de). On the other hand, by using the Skorokhod and minimality conditions on K‾n+1 and C‾n+1 we can show that ⟨Y‾s−n+1,dK‾sn+1⟩≤0 and ⟨Y‾sn+1,dC‾sn+1⟩≤0 . Moreover, from the assumptions (A1.1)–(A1.2), we deduce that for any ε>0 0$]]>, 2⟨Y‾sn+1,f‾Θn(s)⟩≤2|Y‾sn+1|γs|Y‾sn|+κs|Z‾sn|+σs‖U‾sn‖λ≤γs+1ε[κs2+σs2]|Y‾sn+1|2+γs|Y‾sn|2+ε|Z‾sn|2+‖U‾sn‖λ2≤1+1εas2|Y‾sn+1|2+as2|Y‾sn|2+ε|Z‾sn|2+‖U‾sn‖λ2 and |g‾Θn(s)|2≤as2|Y‾sn|2+α|Z‾sn|2+‖U‾sn‖λ2. Plugging these inequalities in (10), and taking the expectation in both side, we deduce that, for any β>0 0$]]> and ε>0 0$]]>, β−1−1εE∫tTeβAsas2|Y‾sn+1|2ds+E∫tTeβAs|Z‾sn+1|2ds+E∫tTeβAs‖U‾sn+1‖λ2ds≤2E∫tTeβAsas2|Y‾sn|2ds+(ε+α)E∫tTeβAs|Z‾sn|2ds+E∫tTeβAs‖U‾sn‖λ2ds. Fix ε>0 0$]]> and define c‾=2/(ε+α) and β0=1+c‾+1/ε . Choosing β≥β0 , we obtain E[c‾∫tTeβAsas2|Y‾sn+1|2ds+∫tTeβAs|Z‾sn+1|2ds+∫tTeβAs‖U‾sn+1‖λ2ds]≤(ε+α)E[c‾∫tTeβAsas2|Y‾sn|2ds+∫tTeβAs|Z‾sn|2ds+∫tTeβAs‖U‾sn‖λ2ds] and by iterations we deduce that c‾Y‾n+1Mβ2,a(Rk)2+Z‾n+1Mβ2(Rk)2+U‾n+1Lβ2(Rk)2≤(ε+α)nc‾Y‾1Mβ2,a(Rk)2+Z‾1Mβ2(Rk)2+U‾1Lβ2(Rk)2. Hence, choosing ε>0 0$]]> such that ε+α<1 , we deduce that (Yn,Zn,Un)n≥1 is a Cauchy sequence in the Banach space Aβ2(Rk) . It remains to show that (Yn)n≥1 is a Cauchy sequence in Sβ2(Rk) . To this end, we define for any integers n,m≥1 ℜn,m=ℜn−ℜm for ℜ∈{Y,Z,U,K,C} , and ∀h∈f,g,hΘn,m(t)=h(t,Θtn)−h(t,Θtm),t≤T. Then it is readily seen that Ytn+1,m+1=∫tTfΘn,m(s)ds+∫tTgΘn,m(s)dBs−∫tTZsn+1,m+1dWs−∫tT∫EUsn+1,m+1(e)μ˜(ds,de)+KTn+1,m+1−Ktn+1,m+1+CT−n+1,m+1−Ct−n+1,m+1. Applying Lemma 2.11 to (11), and taking the essential supremum over τ∈T[0,T] and then the expectation on both sides we get Eess supν∈T[0,T]eβAτ|Yτn+1,m+1|2+βE∫tTeβAsas2|Ysn+1,m+1|2ds+E∫tTeβAs|Zsn+1,m+1|2ds+E∫tTeβAs‖Usn+1,m+1‖λ2ds≤2E∫tTeβAs⟨Ys−n+1,m+1,fΘn,m(s)⟩ds+E∫tTeβAs|gΘn,m(s)|2ds+2Eess supν∈T[0,T]∫0τeβAs⟨Ys−n+1,m+1,Zsn+1,m+1dWs⟩+2Eess supν∈T[0,T]∫0τeβAs⟨Ys−n+1,m+1,gΘn,m(s)dBs⟩+2Eess supν∈T[0,T]∫0τ∫EeβAs⟨Ys−n+1,m+1,Usn+1,m+1(e)μ˜(ds,de)⟩. But, for any ε>0 0$]]>, 2⟨Ysn+1,m+1,fΘn,m(s)⟩≤1εas2|Ysn+1,m+1|2+εfΘn,m(s)as2. Moreover, by the Burkhölder–Davis–Gundy inequality, there exists a universal constant c such that 2Eess supτ∈T[0,T]∫0τeβAs⟨Ys−n+1,m+1,Zsn+1,m+1dWs⟩≤14‖Yn+1,m+1‖Sβ2(Rk)2+4c2‖Zn+1,m+1‖Mβ2(Rk×d)2, 2Eess supτ∈T[0,T]∫0τ∫EeβAs⟨Ys−n+1,m+1,Usn+1,m+1(e)μ˜(ds,de)⟩≤14‖Yn+1,m+1‖Sβ2(Rk)2+4c2‖Un+1,m+1‖Lβ2(Rk)2 and 2Eess supτ∈T[0,T]∫0τeβAs⟨Ys−n+1,m+1,gΘn,m(s)dBs⟩≤14‖Yn+1,m+1‖Sβ2(Rk)2+4c2‖gΘn,m‖Mβ2(Rk×ℓ)2. Hence, there exists C>0 0$]]> such that Eess supν∈T[0,T]eβAτ|Yτn+1,m+1|2≤CE∫0TeβAsfΘn,m(s)as2ds+E∫0TeβAs|gΘn,m(s)|2ds≤C4E∫0TeβAsas2|Ysn,m|2ds+(3+α)E∫0TeβAs|Zsn,m|2ds+E∫0TeβAs‖Usn,m‖λ2ds. Since (Yn,Zn,Un)n≥1 is a Cauchy sequence in Aβ2(Rk) , we deduce that (Yn)n≥1 is a Cauchy sequence in Sβ2(Rk) . Hence, (Yn,Zn,Un)n≥1 is a Cauchy sequence in the Banach space Bβ2(Rk) , so it converges in Bβ2(Rk) to a limit Θ=(Y,Z,U) . Now let us show that (Y,Z,U) , with the additional Mertens process (K,C) , is a solution to RBDSDEJ (2).

Since (Yn,Zn,Un)n≥1 converges in Bβ2(Rk) to a limit (Y,Z,U) , we have limn→+∞(Yn−Y,Zn−Z,Un−U)Bβ2(Rk)2=0. Using the Cauchy–Schwarz inequality and (12), we deduce from (A1.1) and (A1.2) E(|∫tT(f(s,Θsn)−f(s,Θs))ds|2)≤E(1β∫tTeβAs|f(s,Ysn,Zsn,Usn)−f(s,Ys,Zs,Us)|2as2ds)≤3βE(∫tTeβAsas2|Ysn−Ys|2ds+∫tTeβAs|Zsn−Zs|2ds+∫tTeβAs‖Usn−Us‖λ2ds) →n→+∞0. Similarly, by the Burkhölder–Davis–Gundy inequality and (12), we have E(sup0≤t≤T|∫tTg(s,Θsn)dBs−∫tTg(s,Θs)dBs|2)≤E∫tT|g(s,Ysn,Zsn,Usn)−g(s,Ys,Zs,Us)|2ds≤E(∫tTeβAsas2|Ysn−Ys|2ds+α∫tTeβAs|Zsn−Zs|2ds+α∫tTeβAs‖Usn−Us‖λ2ds) →n→+∞0. Moreover, since As≥0 for all s≤T , we have Esup0≤t≤T∫tTZsndWs−∫tTZsdWs2≤E∫tTeβAs|Zsn−Zs|2ds →n→+∞0 and E(sup0≤t≤T|∫tT∫EUsn(e)μ˜(de,ds)−∫tT∫EUs(e)μ˜(de,ds)|2)≤E∫tTeβAs‖Usn−Us‖λ2ds →n→+∞0. For each τ∈T[0,T] , let K˜τ=Kτ−Cτ−=Y0−Yτ−∫0τfs,Θsds−∫0τgs,ΘsdBs+∫0τZsdWs+∫0τ∫EUs(e)μ˜(ds,de). Then, we can easily show that ‖K˜n−K˜‖S22⟶0,asn⟶+∞ . So, letting n⟶+∞ in (9), we deduce that (Y,Z,U,K,C) is a solution to RBDSDEJ (2).

(ii) Uniqueness. Let (Y1,Z1,U1,K1,C1) and (Y2,Z2,U2,K2,C2) be two solutions to RBDSDEJ (2). We define ℜ‾=ℜ1−ℜ2 for ℜ∈{Y,Z,U,K,C} and ∀h∈f,g,h‾Θ(t)=h(t,Θt1)−h(t,Θt2),t≤T. Thus the process (Y‾,Z‾,U‾,K‾,C‾) satisfies the following equation Y‾t=∫tTf‾Θ(s)ds+∫tTg‾Θ(s)dBs−∫tTZ‾sdWs+∫tT∫EU‾s(e)μ˜(ds,de)+K‾T−K‾t+C‾T−−C‾t−. Applying Lemma 2.11 to (13) and taking into consideration Remark 2.3, we have EeβAt|Y‾t|2+βE∫tTeβAsas2|Y‾s|2ds+E∫tTeβAs(|Z‾s|2+‖U‾s‖λ2)ds≤2E∫tTeβAs⟨Y‾s,f‾Θ(s)⟩ds+E∫tTeβAs|g‾Θ(s)|2ds. By the same computations as before (by using the assumptions (A1.1)–(A1.2)), we have, for any ε>0 0$]]>, 2⟨Y‾s,f‾Θ(s)⟩≤2+2εas2|Y‾s|2+ε(|Z‾s|2+|U‾s|2), and |g‾Θ(s)|2≤as2|Y‾s|2+α(|Z‾s|2+|U‾s|2). Hence, choosing ε>0,β>0 0,\hspace{0.2778em}\beta >0$]]> such that ε+α<1 and β>3+2/ε 3+2/\varepsilon $]]>, we deduce that EeβAt|Y‾t|2+β−3−2εE∫tTeβAsas2|Y‾s|2ds+(1−ε−α)E∫tTeβAs|Z‾s|2ds+∫tTeβAs‖U‾s‖λ2ds≤0. It follows that (Y‾,Z‾,U‾)=(0,0,0) , and thus (K‾,C‾)=(0,0) .  □

 Comparison theorem

In all what follows, we are interested in one-dimensional RBDSDEJs (i.e. k=1 ). We consider the RBDSDEJs associated with parameters (fi(.,Θ),g(.,Θ),ξi) for i=1,2 where Θi stands for the process (Yi,Zi,Ui) . Let us state the following assumption (A1.6):ξt1≤ξt2a.s.∀t≤Tf1(t,y,z,u)≤f2(t,y,z,u)a.s.∀(t,y,z,u)∈[0,T]×R×Rd×Lλ. Then we have the following comparison result.

Let (Yi,Zi,Ui,Ki,Ci) be a solution to RBDSDEJs associated with parameters (fi(.,Θ),g(.,Θ),ξi) for i=1,2 . Under the assumptions (A1.1)–(A1.4) and (A1.6) we have ∀t≤T,Yt1≤Yt2,P-a.s.

Define ℜˆ=ℜ1−ℜ2 for ℜ∈{Y,Z,U,K,C,ξ} . Then the process (Yˆ,Zˆ,Uˆ,Kˆ,Cˆ) satisfies the following equation Yˆt=ξˆT+∫tT[f1(s,Θs1)−f2(s,Θs2)]ds+∫tT[g(s,Θs1)−g(s,Θs2)]dBs−∫tTZˆsdWs−∫tT∫EUˆs(e)μ˜(ds,de)+KˆT−Kˆt+CˆT−−Cˆt−. Applying Lemma 2.11, taking into account Remark 2.3 and taking the expectation, we obtain, for all t≤T , EeβAt|Yˆt+|2+βE∫tT1Yˆs>0eβAsas2|Yˆs|2ds+E∫tT1Yˆs>0eβAs|Zˆs|2ds+E∫tT1Yˆs>0eβAs‖Uˆs‖λ2ds≤EeβAT|ξˆT+|2+2E∫tT1Yˆs>0eβAsYˆs+[f1(s,Θs1)−f2(s,Θs2)]ds+E∫tT1Yˆs>0eβAs|g(s,Θs1)−g(s,Θs2)|2ds. 0\right\}}}{e^{\beta {A_{s}}}}{a_{s}^{2}}|{\widehat{Y}_{s}}{|^{2}}ds\\ {} & & \displaystyle +\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}|{\widehat{Z}_{s}}{|^{2}}ds+\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}\| {\widehat{U}_{s}}{\| _{\lambda }^{2}}ds\\ {} & \displaystyle \le & \displaystyle \mathbf{E}\left[{e^{\beta {A_{T}}}}|{\widehat{\xi }_{T}^{+}}{|^{2}}\right]+2\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}{\widehat{Y}_{s}^{+}}[{f^{1}}(s,{\Theta _{s}^{1}})-{f^{2}}(s,{\Theta _{s}^{2}})]ds\\ {} & & \displaystyle +\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}|g(s,{\Theta _{s}^{1}})-g(s,{\Theta _{s}^{2}}){|^{2}}ds.\end{array}\]]]> By assumption (A1.6), we have E[eβAT|ξˆT+|]=0 and Yˆs+[f1(s,Θs1)−f2(s,Θs1)]≤0 , and due to the assumptions (A1.1)–(A1.2), we get, for any ε>0 0$]]>, 2E∫tT1Yˆs>0eβAsYˆs+[f1(s,Θs1)−f2(s,Θs2)]ds≤2E∫tT1Yˆs>0eβAsYˆs+[f2(s,Θs1)−f2(s,Θs2)]ds≤2+2εE∫tT1Yˆs>0eβAsas2|Yˆs+|2ds+εE∫tT1Yˆs>0eβAs|Zˆs|2ds+εE∫tT1Yˆs>0eβAs‖Uˆs‖λ2ds 0\right\}}}{e^{\beta {A_{s}}}}{\widehat{Y}_{s}^{+}}[{f^{1}}(s,{\Theta _{s}^{1}})-{f^{2}}(s,{\Theta _{s}^{2}})]ds\\ {} & \displaystyle \le & \displaystyle 2\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}{\widehat{Y}_{s}^{+}}[{f^{2}}(s,{\Theta _{s}^{1}})-{f^{2}}(s,{\Theta _{s}^{2}})]ds\\ {} & \displaystyle \le & \displaystyle \left(2+\frac{2}{\varepsilon }\right)\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}{a_{s}^{2}}|{\widehat{Y}_{s}^{+}}{|^{2}}ds+\varepsilon \mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}|{\widehat{Z}_{s}}{|^{2}}ds\\ {} & & \displaystyle +\varepsilon \mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}\| {\widehat{U}_{s}}{\| _{\lambda }^{2}}ds\end{array}\]]]> and E∫tT1Yˆs>0eβAs|g(s,Θs1)−g(s,Θs2)|2ds≤E∫tT1Yˆs>0eβAsas2|Yˆs|2ds+αE∫tT1Yˆs>0eβAs(|Zˆs|2+‖Uˆs‖λ2)ds. 0\right\}}}{e^{\beta {A_{s}}}}|g(s,{\Theta _{s}^{1}})-g(s,{\Theta _{s}^{2}}){|^{2}}ds\\ {} & \displaystyle \le & \displaystyle \mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}{a_{s}^{2}}|{\widehat{Y}_{s}}{|^{2}}ds+\alpha \mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}(|{\widehat{Z}_{s}}{|^{2}}+\| {\widehat{U}_{s}}{\| _{\lambda }^{2}})ds.\end{array}\]]]> Plugging these two last inequalities in (14), we deduce that, for any β>0 0$]]> and ε>0 0$]]>, EeβAt|Yˆt+|2+βE∫tT1Yˆs>0eβAsas2|Yˆs|2ds+E∫tT1Yˆs>0eβAs|Zˆs|2ds+E∫tT1Yˆs>0eβAs‖Uˆs‖λ2ds≤3+2εE∫tT1Yˆs>0eβAsas2|Yˆs+|2ds+(ε+α)E∫tT1Yˆs>0eβAs|Zˆs|2ds+E∫tT1Yˆs>0eβAs‖Uˆs‖λ2ds. 0\right\}}}{e^{\beta {A_{s}}}}{a_{s}^{2}}|{\widehat{Y}_{s}}{|^{2}}ds\\ {} & & \displaystyle +\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}|{\widehat{Z}_{s}}{|^{2}}ds+\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}\| {\widehat{U}_{s}}{\| _{\lambda }^{2}}ds\\ {} & \displaystyle \le & \displaystyle \left(3+\frac{2}{\varepsilon }\right)\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}{a_{s}^{2}}|{\widehat{Y}_{s}^{+}}{|^{2}}ds\\ {} & & \displaystyle +(\varepsilon +\alpha )\left(\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}|{\widehat{Z}_{s}}{|^{2}}ds+\mathbf{E}{\int _{t}^{T}}{\mathbb{1}_{\left\{{\widehat{Y}_{s}}>0\right\}}}{e^{\beta {A_{s}}}}\| {\widehat{U}_{s}}{\| _{\lambda }^{2}}ds\right).\end{array}\]]]> Choosing ε=(1−α)/2 and taking β>3+2/ε 3+2/\varepsilon $]]>, we derive that |Yˆt+|2=0a.s.∀t≤T,i.e.,Yt1≤Yt2a.s.∀t≤T.  □

 Reflected BDSDEJs with stochastic growth condition

In this section we are interested in weakening the conditions on the coefficient f. We are also interested in one-dimensional RBDSDEJs (i.e. k=1 ). Let us state the new working assumptions.

Assumptions

We assume that the data (f,g,ξ) satisfy the following assumptions (A2):

There exist four non-negative FtW -measurable processes (γt)t≤T , (κt)t≤T , (σt)t≤T and (ϱt)t≤T such that the condition (A1.2) holds, and there exists another Ft -progressively measurable nonnegative process (ζt)t≤T such that ζa∈Mβ2(R) and for all (t,y,z,u)∈[0,T]×R×Rd×Lλ , |f(t,y,z,u)|≤ζt+γt|y|+κt|z|+σt‖u‖λ.

f(ω,t,·,·,·):R×Rd×Lλ→R is continuous.

The coefficient g satisfies (A1.1) for α∈]0,1/2[ .

The irregular barrier (ξt)t≤T satisfies (A1.4).

Existence of a minimal solution

In this section, we will prove the existence of a minimal solution to RBDSDEJ (2) under the conditions (A2). First let us define a minimal solution as follows.

A solution (Y,Z,U,K,C) to RBDSDEJ (2) is called a minimal solution if for any other solution (Y∗,Z∗,U∗,K∗,C∗) to (2) we have, for each t≤T , Yt≤Yt∗ .

For fixed (ω,t) in Ω×[0,T] , we define the sequence fn(t,y,z,u) associated to the coefficient f as follows: for all (y,y′)∈R2 , (z,z′)∈Rd×Rd and (u,u′)∈Lλ×Lλ , fn(t,y,z,u)=infy′,z′,u′[f(t,y′,z′,u′)+n(γt|y−y′|+κt|z−z′|+σt‖u−u′‖λ)]. From Proposition 4.2 in [30], the sequence fn is well defined for each n≥1 , and it satisfies:

Linear growth condition: ∀n≥1 , ∀(y,z,u)∈R×Rd×Lλ,|fn(t,y,z,u)|≤ζt+γt|y|+κt|z|+σt‖u‖λ.

Monotonicity: ∀(y,z,u)∈R×Rd×Lλ , fn(t,y,z,u) increases in n.

Convergence: If (yn,zn,un)→(y,z,u) in R×Rd×Lλ as n→+∞ , then fn(t,yn,zn,un) →n→+∞f(t,y,z,u).

Lipschitz condition: ∀n≥1 , and for all (y,y′)∈R2 , (z,z′)∈Rd×Rd and (u,u′)∈Lλ×Lλ , we have |fn(t,y,z,u)−fn(t,y′,z′,u′)|≤nγt|y−y′|+nκt|z−z′|+nσt‖u−u′‖λ.

We also define the function F(t,y,z,u)=ζt+γt|y|+κt|z|+σt‖u‖λ. Now, from Theorem 3.4, there exist two processes Θ‾:=(Y‾,Z‾,U‾) and Θn:=(Yn,Zn,Un) which are the solutions to RBDSDEJs associated with parameters (F(.,Θ‾),g(.,Θ‾),ξ) and (fn(.,Θn),g(.,Θn),ξ) , respectively.

From the definitions of fn and F together with (15), we observe that ∀n≥1 , fn≤fn+1≤F . Then, due to Theorem 3.5 we have ∀t≤T,Yt1≤Ytn≤Ytn+1≤Y‾t.

The proof of the main result of this section is based on the two next lemmas.

Under the assumption (A2), there exists a positive constant Λ depending on β such that (Y‾,Z‾,U‾)Bβ2(R)2≤Λ‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2 and for each n≥1 (Yn,Zn,Un)Bβ2(R)2≤Λ‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2.

We know that Y‾t=ξT+∫tTF(s,Θ‾s)ds+∫tTg(s,Θ‾s)dBs−∫tTZ‾sdWs−∫tT∫EU‾s(e)μ˜(ds,de)+K‾T−K‾t+C‾T−−C‾t−, where (K‾,C‾) satisfies the Skorokhod and minimality conditions. Then, applying Lemma 2.11 together with Remark 2.3, we deduce eβAt|Y‾t|2+β∫tTeβAsas2|Y‾s|2ds+∫tTeβAs|Z‾s|2ds+∫tTeβAs‖U‾s‖λ2ds≤eβAT|ξ|2+2∫tTeβAsY‾sF(s,Θ‾s)ds+2∫tTeβAsY‾s−g(s,Θ‾s)dBs−2∫tTeβAsY‾s−Z‾sdWs−2∫tT∫EeβAsY‾s−U‾s(e)μ˜(ds,de)+∫tTeβAs|g(s,Θ‾s)|2ds+2∫tTeβAsY‾s−dK‾s+2∫tTeβAsY‾sdC‾s. But for any β>0 0$]]> and ε>0 0$]]>, 2Y‾sF(s,Θ‾s)≤β2+2+2εas2|Y‾s|2+2βζsas2+ε(|Z‾s|2+‖U‾s‖λ2) and |g(s,Θ‾s)|2≤2|g(s,Θ‾s)−g(s,0)|2+|g(s,0)|2≤2as2|Y‾s|2+α(|Z‾s|2+‖U‾s‖λ2)+|g(s,0)|2. Plugging these inequalities in (19) and taking expectation, we obtain β2−4−2εE∫tTeβAsas2|Y‾s|2ds+(1−ε−2α)E∫tTeβAs|Z‾s|2ds+(1−ε−2α)E∫tTeβAs‖U‾s‖λ2ds≤EeβAT|ξ|2+2β∫tTeβAsζsas2ds+2∫tTeβAs|g(s,0)|2ds+2∫tTeβAsY‾s−dK‾s+2∫tTeβAsY‾sdC‾s. Moreover, 2E∫tTeβAsY‾s−dK‾s+2E∫tTeβAsY‾sdC‾s≤2Eess supτ∈T[0,T]eβAτ|ξτ|∫0Td(K‾t+C‾t)≤εEess supτ∈T[0,T]e2βAτ|ξτ|2+1εE(K‾T+C‾T)2, and from (18) we have E(K‾T+C‾T)2≤6EY‾02+ξT2+∫0TF(s,Θ‾s)ds2+∫0Tg(s,Θ‾s)dBs2+∫0TZ‾sdWs2+∫0T∫EU‾s(e)μ˜(ds,de)2≤6EY‾02+ξT2+1β∫0TeβAsF(s,Θ‾s)as2ds+c∫0T|g(s,Θ‾s)|2ds+∫0T|Z‾s|2ds+∫0T‖U‾s‖λ2ds≤6E2ess supτ∈T[0,T]e2βAτ|ξτ|2+4β∫0TeβAsζsas2ds+2c∫0TeβAs|g(s,0)|2ds+4β+2c∫0TeβAsas2|Y‾s|2ds+4β+2αc+c∫0TeβAs|Z‾s|2ds+∫0TeβAs‖U‾s‖λ2ds. Then (20) becomes ϕ1E∫tTeβAsas2|Y‾s|2ds+ϕ2E∫tTeβAs|Z‾s|2ds+ϕ2E∫tTeβAs‖U‾s‖λ2ds≤Λ1Eess supτ∈T[0,T]e2βAτ|ξτ|2+∫0TeβAsζsas2ds+∫0TeβAs|g(s,0)|2ds. where ϕ1=β2−4−2ε−6ε(4β+2c) , ϕ2=1−ε−2α−6ε(4β+2αc+c) and Λ1 is a nonnegative constant depending on β, c and ε. Now, choose ε≤1−2α with 0<α<1/2 and β>0 0$]]> such that εβ(β−12−24c)>48 48$]]> (these choices are suitable to obtain a nonnegative ϕ1 and ϕ2 ). Hence Y‾,Z‾,U‾Aβ2(R)2≤Λ1‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2. To conclude, we need an estimate of Y‾Sβ2(R)2 . For this, using (19) once again and (21), we have Eess supτ∈T[0,T]eβAτ|Y‾τ|2≤Λ1‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2+2Eess supτ∈T[0,T]∫0τeβAsY‾s−g(s,Θ‾s)dBs+2Eess supτ∈T[0,T]∫0τeβAsY‾s−Z‾sdWs+2Eess supτ∈T[0,T]∫0τ∫EeβAsY‾s−U‾s(e)μ˜(ds,de). By the Burkhölder–Davis–Gundy inequality, there exists c>0 0$]]> such that 2Eess supτ∈T[0,T]∫0τeβAsY‾s−Z‾sdWs≤16Y‾Sβ2(R)2+6c2Z‾Mβ2(Rd)2, 2Eess supτ∈T[0,T]∫0τeβAsY‾s−g(s,Θ‾s)dBs≤16Y‾Sβ2(R)2+6c2g(.,Θ‾)Mβ2(Rℓ)2≤16Y‾Sβ2(R)2+12c2Y‾Mβ2,a(R)2+αZ‾Mβ2(Rd)2+αU‾Lβ2(R)2+g(.,0)Mβ2(Rℓ)2 and 2Eess supτ∈T[0,T]∫0τ∫EeβAsY‾s−U‾s(e)μ˜(ds,de)≤16Y‾Sβ2(R)2+6c2U‾Lβ2(R)2. Then, we derive that Y‾Sβ2(R)2≤Λ2‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2 where Λ2 is a nonnegative constant depending on β, c and ε. The desired result is obtained by combining the estimates (21) and (22) with Λ=Λ1∨Λ2 . As a consequence, from (17) we deduce that YnSβ2(R)2≤Λ2‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2. Using the same computations as before, we can prove that Yn,Zn,UnAβ2(R)2≤Λ1‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2.  □

Under the assumption (A2) the sequence of processes (Yn,Zn,Un)n≥1 converges almost surely in Bβ2(R) for each β>2 2$]]>.

We know that Ytn=ξT+∫tTfn(s,Θsn)ds+∫tTg(s,Θsn)dBs−∫tTZsndWs−∫tT∫EUsn(e)μ˜(ds,de)+KTn−Ktn+CT−n−Ct−n, where (Kn,Cn) satisfy the Skorokhod and minimality conditions. We define, for any integers n,m≥1 , ℜn,m=ℜn−ℜm for ℜ∈{Y,Z,U,K,C} , Δfn,m(t)=fn(t,Θtn)−fm(t,Θtm)andΔgn,m(t)=g(t,Θtn)−g(t,Θtm),t≤T. Then, applying Lemma 2.11 together with Remark 2.3, we get βE∫tTeβAsas2|Ysn,m|2ds+E∫tTeβAs|Zsn,m|2ds+E∫tTeβAs‖Usn,m‖λ2ds≤E∫tTeβAs|Δgn,m(s)|2ds+2E∫tTeβAsYsn,mΔfn,m(s)ds. Using the assumption (A2.3) and the basic inequality 2ab≤ϵa2+b2ϵ , we get (β−1)Yn,mMβ2,a(R)2+(1−α)Zn,mMβ2(Rd)2+Un,mLβ2(R)2≤E∫0TeβAsas2|Ysn,m|2ds+E∫0TeβAsΔfn,m(s)as2ds. Next, from the linear growth condition on fn and fm , and by Lemma 4.1, we find (β−2)Yn,mMβ2,a(R)2+(1−α)Zn,mMβ2(Rd)2+Un,mLβ2(R)2≤8Λ‖ξ‖S2β2(R)2+ζaMβ2(R)2+‖g(.,0)‖Mβ2(Rℓ)2. Hence for β>2 2$]]>, we deduce that (Yn,Zn,Un) is a Cauchy sequence in Aβ2(R) , so it converges in Aβ2(R) . On the other hand, from (17) we deduce that there exists a process Y∈Sβ2(R) such that Yn→Y a.s. as n→∞ . The result follows.  □

The main result in this section is what follows.

Under the assumptions (A2), the RBDSDEJ (2) associated with parameters (f(.,Θ),g(.,Θ),ξ) has a minimal solution (Y,Z,U,K,C)∈Bβ2(R)×S2(R)×S2(R) .

From (17), it is readily seen that (Yn)n≥1 converges to Y a.s. in Sβ2(R) . Otherwise, due to Lemma 4.2 there exist two subsequences still noted as the whole sequences (Zn)n≥1 and (Un)n≥1 such that Θn=(Yn,Zn,Un) converges to Θ=(Y,Z,U)∈Aβ2(R) as n→+∞ . By (16), we have fn(t,Θtn) →n→+∞f(t,Θt),t≤T. Furthermore, using the linear growth condition of fn , it follows that E∫0TeβAsfn(s,Θsn)as2ds≤4E(∫0TeβAsζsas2ds+supn∫0TeβAsas2|Ysn|2ds+supn∫0TeβAs|Zsn|2ds+supn∫0TeβAs‖Usn‖λ2ds), and by Lemma 4.1 we deduce that E∫0TeβAsfn(s,Θsn)as2ds≤4Λ‖ξ‖S2β2(R)2+(1+Λ)ζaMβ2(R)2+Λ‖g(.,0)‖Mβ2(Rℓ)2. Since E∫0Tfn(s,Θsn)ds2≤1βE∫0TeβAsfn(s,Θsn)as2ds, by Lebesgue’s dominated convergence theorem, we deduce that, for almost all t≤T , E∫0T(fn(s,Θsn)−f(s,Θs))ds2 →n→+∞0. We have also, for almost all t≤T , E∫0T(g(s,Θsn)−g(s,Θs))dBs2 →n→+∞0. Moreover, we have Esup0≤t≤T∫tTZsndWs−∫tTZsdWs2≤E∫tTeβAs|Zsn−Zs|2ds →n→+∞0 and E(sup0≤t≤T|∫tT∫EUsn(e)μ˜(de,ds)−∫tT∫EUs(e)μ˜(de,ds)|2)≤E∫tTeβAs‖Usn−Us‖λ2ds →n→+∞0. Next, for each τ∈T[0,T] , let K˜τ=Kτ−Cτ−=Y0−Yτ−∫0τfs,Θsds−∫0τgs,ΘsdBs+∫0τZsdWs+∫0τ∫EUs(e)μ˜(ds,de). Then, we can easy show that ‖K˜n−K˜‖S22⟶0,asn⟶+∞ . So, letting n⟶+∞ in (23), we deduce that (Y,Z,U,K,C) is a solution to RBDSDEJ (2).

Now, let (Y∗,Z∗,U∗,K∗,C∗)∈Bβ2(R)×S2(R)×S2(R) be another solution to RBDSDEJ (2). By virtue of Theorem 3.5, we deduce that ∀n≥1,Yn≤Y∗. Therefore, by passing to the limit Y≤Y∗ one proves that Y is the minimal solution to RBDSDEJ (2).  □

 Acknowledgments

The authors are greatly grateful to the editor and the referee for the careful reading and many constructive suggestions, which significantly contributed to improving the quality of the paper.

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