Reflected generalized backward stochastic differential equations (BSDEs) with one discontinuous barrier are investigated when the noise is driven by a Brownian motion and an independent Poisson measure. The existence and uniqueness of the solution are derived when the generators are monotone and the barrier is right-continuous with left limits (rcll). The link is established between this solution and a viscosity solution for an obstacle problem of integral-partial differential equations with nonlinear Neumann boundary conditions.
Generalized BSDE with jumpsreflected BSDErcll barrierviscosity solutionintegral-partial differential equations60H0560H1060H3035D40Introduction
General nonlinear backward stochastic differential equations (BSDEs, for short) in the framework of Brownian motion were first introduced by Pardoux and Peng [18] and then extended to the case of jumps by Tang and Li [28] and Rong [26]. Since then, the theory of BSDEs has grown rapidly and has been applied to various areas such as mathematical finance [4, 5], stochastic control and stochastic game theory [13, 16] and the theory of partial differential equations [1, 19, 20, 22].
BSDEs have been extended to the reflected case by El Karoui et al. [6] in the case of a Brownian filtration. In their setting, one of the components of the solution is forced to stay above a given barrier which is a continuous adapted stochastic process. Hamadène and Ouknine [11] treat the case where the filtration is generated by the Brownian motion and a Poisson random measure when the square-integrable obstacle has only inaccessible jump times. Later on, Essaky [9] and Hamadène and Ouknine [12] have extended these results to a right-continuous left-limited (rcll) obstacle with predictable and inaccessible jumps. In [8], one of the authors studied the reflected BSDE driven by a Lévy process in both cited cases.
In another context, in order to provide a probabilistic representation for a solution of a system of parabolic or elliptic semilinear PDEs with nonlinear Neumann boundary condition, Pardoux and Zhang in [21] initiated a new class of BSDEs which involves the integral with respect to a continuously increasing process, which is the local times of a diffusion process on the boundary. This kind of equations is called generalized BSDEs. Following this way, Pardoux in [17] considered the generalized BSDE with jumps, El Otmani in [7] studied the generalized BSDE driven by Lévy process, Ren and Xia [25] have introduced the notion of reflected generalized BSDEs and Ren and El Otmani [24] have treated the reflected generalized BSDE driven by Lévy process.
In this paper, we focus on the following reflected generalized BSDEs with jumps and rcll barrier (reflected generalized BSDEs, for short)
Yt=ξ+∫tTf(s,Ys,Zs,Vs)ds+∫tTg(s,Ys)dAs+KT−Kt−∫tTZsdWs−∫tT∫UVs(e)N˜(ds,de),Yt≥Lt,t≤Tand∫0T(Yt−−Lt−)dKt=0,
where W is a standard Brownian motion and N˜ is a compensated Poisson random measure. The second condition in (1) indicates that the first component Y of the solution is forced to stay above the barrier L and the role of K is to push Y upwards in order to keep it above L in a minimal way according to the second part of this condition, i.e. K increases only when Y=L. Note that in this case, the jumps can be inaccessible (which stem from the stochastic integral with respect to N˜) or predictable (which comes from the predictable negative jumps of the barrier L) (see, e.g., [3]).
Due to the presence of the process (Kt)t≤T, the integral with respect to the continuous increasing process (At)t≤T, which is the local time of a solution for a reflected stochastic differential equation with jumps, and the following special form of the coefficient:
f(t,x,y,z,r)=f(t,x,y,z,∫Ur(e)γ(x,e)λ(de)),
we will establish that equation (1) provides, in a Markovian case, a probabilistic interpretation of the solution, in the viscosity sense, of the following obstacle problem of parabolic integral-partial differential equation (IPDE) with nonlinear Neumann boundary conditions
(u−ℓ)∧(−∂u∂t−Lu−f(t,x,u,(∇xuσ),Bu))=0,∀(t,x)∈[0,T]×G;u(T,x)=H(x),∀x∈G,∂u∂n+g(t,x,u)=0,∀x∈∂G,
where:
G is an open connected bounded domain of Rl(l≥1) which is such that for a function Φ∈Cb2(R), G and its boundary ∂G are characterized by G={Φ>0}0\}$]]>, ∂G={Φ=0} and for any x∈∂G, ∇Φ(x) is the unit normal vector pointing toward the interior of G.
L is the second-order integral-differential operator
L=R+S
with
Rϕ=12Tr[σσT(x)]Dx2ϕ(t,x)+⟨b(x),∇xϕ(t,x)⟩,Sϕ=∫U(ϕ(t,x+c(x,e))−ϕ(t,x)−⟨∇xϕ(t,x),c(x,e)⟩)λ(de).
B is an integral operator defined as
Bϕ=∫U(ϕ(t,x+c(x,e))−ϕ(t,x))γ(x,e)λ(de).
For every x∈∂G,
∂ϕ∂n=⟨∇xϕ,∇Φ(x)⟩.
f, g, H, ℓ, b, σ, c and γ are supposed to satisfy suitable assumptions.
Therefore, in the first part of this paper, the main objective is to prove the uniqueness and existence of the solution of (1) when the coefficients f and g are only monotone w.r.t. y and satisfy a linear growth condition. We first consider the case when the coefficients f and g are Lipschitz, then we solve the problem when f and g depend only on (t,y) and we generalize the result using the fixed point theorem.
The second main aim of this paper is to deal with the obstacle problem of the IPDE with nonlinear Neumann boundary condition (2). By using the results obtained in the first part, i.e. related to the existence and uniqueness of the solution of (1) we prove that equation (2) has a unique viscosity solution.
The paper is organized as follows. Section 2 is devoted to the study of the reflected generalized BSDE: first, we establish a priori estimate of the solution, and then we prove the uniqueness and existence of the solution. A comparison theorem will be presented. Section 3 focuses on the link between this reflected generalized BSDE and the obstacle problem of IPDE with nonlinear Neumann boundary conditions.
Preliminaries and notations
Let T>00$]]> be a fixed time and consider a probability space (Ω,F,P) carrying a standard d-dimensional Brownian motion (Wt)t≤T and an independent martingale measure (N˜t)t≤T corresponding to a standard Poisson random measure N on R+×U where U:=Rk∖{0}(k≥1) is equipped with its Borel σ-algebra U. Namely, for any Borel measurable subset Λ∈U such that λ(Λ)<∞, it holds N˜t(Λ):=Nt(Λ)−tλ(Λ) where λ is assumed to be a σ-finite measure on (U,U) and satisfying ∫U(1∧|e|2)λ(de)<∞.
We assume that Ft:=FtW∨FtN˜ where FtN˜={∫[0,s]×ΛN(du,de),s≤t,Λ∈U}.
We will denote by |.| the Euclidian norm on Rd and for a given rcll process (Xt)t≤T,
Xt−=lims↗tXsandΔXt=Xt−Xt−,t≤T.
Let (At)t∈[0,T] be a continuous one-dimensional increasing Ft-progressively measurable real valued process satisfying A0=0.
For every μ>00$]]>, we denote:
P (resp. Pd) is the σ-algebra of Ft-progressively measurable (resp. predictable) sets on [0,T]×Ω.
Sμ2 is the space of R-valued rcll Ft-adapted processes (Yt)t≤T such that
‖Y‖Sμ22=E[sup0≤t≤TeμAt|Yt|2]<∞.
Hμ,A2 is the space of R-valued rcll Ft-adapted processes (Yt)t≤T such that
‖Y‖Hμ,A22=E∫0TeμAt|Yt|2dAt<∞.
Hμ2 is the space of Rd-valued P-measurable processes (Zt)t≤T such that
‖Z‖Hμ22=E∫0TeμAt|Zt|2dt<∞.
Lλ2 is the space of R-valued and Pd⊗U-measurable mapping V:Ω×[0,T]×U→R such that
‖V‖λ2=∫U|V(e)|2λ(de)<∞.
Lμ,λ2 is the subspace of Lλ2 which contains the mapping V(t,ω,e) such that
‖V‖Lμ,λ22=E∫0TeμAt‖Vt‖λ2dt<∞.
K2 is the subspace of the Ft-predictable, rcll and nondecreasing processes (Kt)t≤T such that K0=0 and E|KT|2<∞.
Lμ2:=(Sμ2∩Hμ,A2)×Hμ2×Lμ,λ2 and Dμ2=Lμ2×Sμ2.
We consider the data (ξ,f,g,L) composed by:
A terminal value ξ which is a FT-measurable variable such that
E[eμAT|ξ|2]<∞.
Two functions f:[0,T]×Ω×R×Rd×Lλ2⟶R and g:[0,T]×Ω×R⟶R such that, for some κ>00$]]>, α∈R and β<0, for all t∈[0,T] and (y,z,v),(y′,z′,v′)∈R×R×Lλ2:
(y−y′)(f(t,y,z,v)−f(t,y′,z,v))≤α|y−y′|2,
|f(t,y,z,v)−f(t,y,z′,v′)|≤κ(|z−z′|+‖v−v′‖λ),
(y−y′)(g(t,y)−g(t,y′))≤β|y−y′|2,
|f(t,y,0,0)|≤φt+κ|y| and |g(t,y)|≤ψt+κ|y| where φ and ψ are two adapted processes with values in [1,+∞[ such that
E∫0TeμAt|φt|2dt+E∫0TeμAt|ψt|2dAt<+∞,
y↦(f(t,y,z,v),g(t,y)) is continuous for all (z,v), (t,ω) a.s.
(Lt)t≤T is an obstacle which is an Ft-progressively measurable rcll real-valued process satisfying
E[sup0≤t≤T|eμAt(Lt)+|2]<∞andLT≤ξ,P-a.s.
Hypothesis (H2)(iv) implies that
E[eμAT]≤1+μE∫0TeμAt|ψt|2dAt<+∞.
As in [2, p. 1137] we can suppose w.l.o.g. that α=0 and also from [21] we can show that β<0 is not a severe restriction.
Reflected generalized BSDEs with <italic>rcll</italic> barrier
A solution of the reflected generalized BSDE is a quadruplet (Y,Z,V,K) satisfying
(i)E[sup0≤t≤T|Yt|2+∫0T|Yt|2dAt+∫0T(|Zt|2+‖Vt‖λ2)dt]<∞,(ii)Yt=ξ+∫tTf(s,Ys,Zs,Vs)ds+∫tTg(s,Ys)dAs+KT−Kt−∫tTZsdWs−∫tT∫UVs(e)N˜(ds,de),(iii)Yt≥Lt,t≤T,(iv)Kis a nondecreasing rcll process withK0=0andE|KT|2<∞andK=Kc+KdwhereKc(resp.Kd)is the continuous(resp. purely discontinuous) part ofKand almost surely∫0T(Yt−Lt)dKtc=0andΔKtd=(Yt−Lt−)−𝟙{Yt−=Lt−}.
The jump of the process Y can be inaccessible or predictable: the inaccessible jumps come from the martingales (∫0t∫UVs(e)N˜(ds,de))t and the predictable jumps are derived from the negative jumps of the process L.
The Skorokhod condition ∫0T(Yt−−Lt−)dKt=0 in (1) and the characterization (iv) in (3) are equivalent. Indeed, if the Skorokhod condition is satisfied, then
∫0T(Ys−Ls)dKsc=0and∫0T(Ys−−Ls−)dKsd=0.
The process Kd does act only when the process Y has a predictable jump. In this case, the role of Kd is to make the necessary jump to Y in order to bring it above L. Therefore, for every predictable stopping time τ≤T, we have
ΔKτd=−ΔYτ=−(Yτ−Yτ−)=(Lτ−−Yτ)+I(Lτ−=Yτ−)∩(ΔLτ<0)=(Yτ−Lτ−)−I(Lτ−=Yτ−).
Conversely,
∫0T(Yt−−Lt−)dKt=∫0T(Yt−Lt)dKtc+∑0<t≤T(Yt−−Lt−)ΔKtd=∫0T(Yt−Lt)dKtc+∑0<t≤T(Yt−−Lt−)(Yt−Lt−)−𝟙{Yt−=Lt−}=0.
A priori estimate
Assume that (H1)–(H3) hold and let(Y,Z,V,K)and(Y′,Z′,V′,K′)be the solutions of reflected generalized BSDE (3) with data(ξ,f,g,L)and(ξ′,f′,g′,L′), respectively. Then there exists a constantC=C(α,β,μ,T,κ)such thatsup0≤t≤TE[eμAt|Yt−Yt′|2]+E∫0TeμAs[|Ys−Ys′|2dAs+[|Zs−Zs′|2+‖Vs−Vs′‖λ2]ds]≤CE[eμAT|ξ−ξ′|2]+E∫0TeμAs|f(s,Ys′,Zs′,Vs′)−f′(s,Ys′,Zs′,Vs′)|2ds+E∫0TeμAs|g(s,Ys′)−g′(s,Ys′)|2dAs+E∫0TeμAs[(Ls−−Ys−′)dKs−(Ys−−Ls−′)dKs′].
Denote ℜ‾=ℜ−ℜ′ for ℜ=Y,Z,V,K. Using Itô’s formula (see [23, Theorem 33, p. 81]), we can write, for some γ>00$]]> and for all t≤T,
eγT+μAT|ξ‾|2=eγt+μAt|Y‾t|2+γ∫tTeγs+μAs|Y‾s|2ds+μ∫tTeγs+μAs|Y‾s|2dAs−2∫tTeγs+μAsY‾s[f(s,Ys,Zs,Vs)−f′(s,Ys′,Zs′,Vs′)]ds−2∫tTeγs+μAsY‾s[g(s,Ys)−g′(s,Ys′)]dAs−2E∫tTeγs+μAsY‾s−dK‾s+2∫tTeγs+μAsY‾sZ‾sdWs+2∫tT∫Ueγs+μAsY‾s−V‾s(e)N˜(ds,de)+∫tTeγs+μAs|Z‾s|2ds+∫tT∫Ueγs+μAs|V‾s(e)|2N(ds,de).
Taking expectation and using assumption (H2), the Skorokhod condition and the inequality 2|ab|≤εa2+b2ε, we obtain
E[eγt+μAt|Y‾t|2]+E∫tTeγs+μAs|Y‾s|2(γds+μdAs)+E∫tTeγs+μAs(|Z‾s|2+‖V‾s‖λ2)ds≤E[eγT+μAT|ξ‾|2]+(1+2α++4κ2)E∫tTeγs+μAs|Y‾s|2ds+12E∫tTeγs+μAs[|Z‾s|2+‖V‾s‖λ2]ds+12|β|E∫tTeγs+μAs|g(s,Ys′)−g′(s,Ys′)|2dAs+E∫tTeγs+μAs|f(s,Ys′,Zs′,Vs′)−f′(s,Ys′,Zs′,Vs′)|2ds+2E∫tTeγs+μAs(Ls−−Ys−′)dKs−2E∫tTeγs+μAs(Ys−−Ls−′)dKs′.
Choosing γ=1+2α++4κ2, we get the desired result. □
Let(Y,Z,V,K)be the solution of (3). Then there exists a constantC>00$]]>such thatE[sup0≤t≤TeμAt|Yt|2]+E∫0TeμAs|Ys|2dAs+E∫0TeμAs[|Zs|2+‖Vs‖λ2]ds+E|KT|2≤C{E[eμAT|ξ|2]+E∫0TeμAs[|φs|2ds+|ψs|2dAs]+Esup0≤t≤T|eμAt(Lt)+|2}.
Remark that (Y′,Z′,V′,K′)=(0,0,0,0) is the unique solution of (3) with data (ξ′,f′,g′,L′)=(0,0,0,0). Then, directly by Proposition 1 we have, for all t≤T,
sup0≤t≤TE[eμAt|Yt|2]+E∫0TeμAs|Ys|2dAs+E∫0TeμAs[|Zs|2+‖Vs‖λ2]ds≤CE[eμAT|ξ|2+∫0TeμAs|f(s,0,0,0)|2ds+∫0TeμAs|g(s,0)|2dAs+∫0TeμAsLs−dKs]≤CE[eμAT|ξ|2+∫0TeμAs[|φs|2ds+|ψs|2dAs]]+ρC2Esup0≤t≤T|eμAt(Lt)+|2+1ρE|KT|2.
However, by (3)(ii) and (H2) we get by using the Hölder inequality
E|KT|2≤6|Y0|2+E|ξ|2+E(∫0T|f(s,Ys,Zs,Vs)|ds)2+E(∫0T|g(s,Ys)|dAs)2+E|∫0TZsdWs|2+E|∫0T∫UVs(e)N˜(ds,de)|2≤6|Y0|2+6E|ξ|2+24TE∫0T|φs|2ds+12μE∫0TeμAs|ψs|2dAs+24κ2TE∫0T|Ys|2ds+6(1+4κ2T)E∫0T[|Zs|2+‖Vs‖λ2]ds+12κ2μE∫0TeμAs|Ys|2dAs≤6E[eμAT|ξ|2]+24TE∫0TeμAs|φs|2ds+12μE∫0TeμAs|ψs|2dAs+6(1+4κ2T2)sup0≤t≤TE[eμAt|Yt|2]+12κ2μE∫0TeμAs|Ys|2dAs+6(1+4κ2T)E∫0TeμAs[|Zs|2+‖Vs‖λ2]ds.
Choosing ρ>max{6(1+4κ2T2);6(1+4κ2T);12κ2μ}\max \{6(1+4{\kappa ^{2}}{T^{2}});\hspace{2.5pt}6(1+4{\kappa ^{2}}T);\hspace{2.5pt}\frac{12{\kappa ^{2}}}{\mu }\}$]]>, we obtain
sup0≤t≤TE[eμAt|Yt|2]+E∫0TeμAs|Ys|2dAs+E∫0TeμAs[|Zs|2+‖Vs‖λ2]ds≤C1{E[eμAT|ξ|2]+E∫0TeμAs[|φs|2ds+|ψs|2dAs]+Esup0≤t≤T|eμAt(Lt)+|2},
implying that
E|KT|2≤C2E[eμAT|ξ|2]+E∫0TeμAs[|φs|2ds+|ψs|2dAs]+Esup0≤t≤T|eμAt(Lt)+|2.
On the other hand,
E[sup0≤t≤TeμAt|Yt|2]≤E[eμAT|ξ|2]+(1+2κ+2κ2)E∫0TeμAs|Ys|2ds+E∫0TeμAs[|φs|2ds+|ψs|2μdAs]+E|KT|2+Esup0≤t≤T|eμAt(Lt)+|2+2Esup0≤t≤T|∫tTeμAsYsZsdWs|+2Esup0≤t≤T|∫tT∫UeμAsYs−Vs(e)N˜(ds,de)|.
Applying the Burkholder–Davis–Gundy inequality (see, e.g., [23, Theorem 48, p. 195]), there is a universal positive constant c such that
2Esup0≤t≤T|∫tTeμAsYsZsdWs|≤14E[sup0≤t≤TeμAt|Yt|2]+4c2E∫0TeμAs|Zs|2ds
and
2Esup0≤t≤T|∫tT∫UeμAsYs−Vs(e)N˜(ds,de)|≤2cE(∫0T∫Ue2μAs|Ys−|2|Vs(e)|2N(ds,de))12≤14E[sup0≤t≤TeμAt|Yt|2]+4c2E∫0TeμAs|Vs(e)|2N(ds,de)=14E[sup0≤t≤TeμAt|Yt|2]+4c2E∫0TeμAs‖Vs‖λ2ds.
Plugging those inequalities in (4), we conclude that
E[sup0≤t≤TeμAt|Yt|2]≤C3{E[eμAT|ξ|2]+E∫0TeμAs[|φs|2ds+|ψs|2dAs]+Esup0≤t≤T|eμAt(Lt)+|2}.
The result is therefore verified. □
Existence and uniqueness results
Under the hypothesis (H1)–(H3), the reflected generalized BSDE (3) associated with the data(ξ,f,g,L)has at most one solution.
A direct consequence of Proposition 1. □
Our approach to prove existence is based on the following strategy: first, we establish uniqueness and existence when the coefficients f and g are Lipschitz, then we solve the problem when f does not depend on (z,v) and then we generalize the result using the fixed point theorem. The following proposition gives the first step.
Suppose that the assumptions (H1)–(H3) hold forμ>11$]]>. We suppose in addition that, for allt≤Tandy,y′∈R,|f(t,y,z,v)−f(t,y′,z,v)|+|g(t,y)−g(t,y′)|≤κ′|y−y′|.Then the reflected generalized BSDE (3) has a unique solution.
We can use two methods to prove this claim. We have omitted the first technique, which is identical to that used in [9] and is based on a monotonic limit theorem and penalization method. The fixed point argument is utilized in the second one. Here is the proof of that.
First, let us endow the space Lμ2 with the norm
‖(Y,Z,V)‖γ,μ2=E[∫0Teγt+μAt[|Yt|2+|Zt|2+‖Vt‖λ2]dt+∫0Teγt+μAt|Yt|2dAt].
We define the map Ψ of (Lμ2, ‖.‖γ,μ) into itself as follows: for every (Y,Z,V)∈Lμ2, we put
Ψ(Y,Z,V)=(Y˜,Z˜,V˜),
where (Y˜,Z˜,V˜,K˜) is the solution of the reflected generalized BSDE associated with (ξ,f(t,Y,Z,V),g(t,Y),L).
The process (Y˜,Z˜,V˜,K˜) is constructed as follows. Let η be the process defined by
ηt=ξI{t=T}+LtI{t<T}+∫0tf(s,Ys,Zs,Vs)ds+∫0tg(s,Ys)dAs.
Note that η is rcll and E(sup0≤t≤T|ηt|2)<∞. Let S(η) be the Snell envelope of η given by
St(η)=ess supν∈TtE[ην|Ft].
However, it is the smallest rcll supermartingale dominating the process η which verifies
Esup0≤t≤T|St(η)|2<∞.
Then, S(η) is of class [D]. Henceforth, it has the following Doob–Meyer decomposition (see [23, Theorem 8, p. 111]):
St(η)=E[ξ+∫0Tf(s,Ys,Zs,Vs)ds+∫0Tg(s,Ys)dAs+K˜T|Ft]−K˜t
where K˜ is an Ft-adapted rcll nondecreasing process (K˜0=0) and E|K˜T|2<∞. Through the martingale representation theorem, there exists two processes Z˜ and V˜ such that
ξ+∫0Tf(s,Ys,Zs,Vs)ds+∫0Tg(s,Ys)dAs+K˜T=E[ξ+∫0Tf(s,Ys,Zs,Vs)ds+∫0Tg(s,Ys)dAs+K˜T|Ft]+∫0TZ˜sdWs+∫0T∫UV˜s(e)N˜(ds,de),
where E∫0T|Z˜s|2ds<∞ and E∫0T‖V˜s‖λ2ds<∞.
Then, if we denote
Y˜t=ess supτ∈TtE[ξI(τ=T)+LτI(τ<T)+∫tτf(s,Ys,Zs,Vs)ds+∫tτg(s,Ys)dAs|Ft],
we get
Y˜t+∫0tf(s,Ys,Zs,Vs)ds+∫0tg(s,Ys)dAs+K˜t=St(η)+K˜t=ξ+∫0Tf(s,Ys,Zs,Vs)ds+∫0Tg(s,Ys)dAs+K˜T−∫tTZ˜sdWs−∫tT∫UV˜s(e)N˜(ds,de).
Hence
Y˜t=ξ+∫tTf(s,Ys,Zs,Vs)ds+∫tTg(s,Ys)dAs+K˜T−K˜t−∫tTZ˜sdWs−∫tT∫UV˜s(e)N˜(ds,de).
To finish this construction it remains to show that
∫0T(Y˜t−−Lt−)dK˜td=∫0T(Y˜t−Lt)dK˜tc=0.
First, recall that {ΔK˜d>0}⊂{S−(η)=η_}0\}\subset \{{\mathcal{S}_{-}}(\eta )=\underline{\eta }\}$]]> where η_s=lim supt↗sηt (see, e.g., [6, Proposition 2.34, p. 131]). So, we can write
∫0T(Y˜s−−Ls−)dK˜sd=∑0<s≤T(Y˜s−−Ls−)I{ΔK˜sd>0}ΔK˜sd=∑0<s≤T(Y˜s−−Ls−)(ηs−−Ss(η))+I{ηs−=Ss−(η)}=0.0\}}}\Delta {\tilde{K}_{s}^{d}}\\ {} & =\sum \limits_{0 < s\le T}({\tilde{Y}_{s-}}-{L_{s-}}){\big({\eta _{s-}}-{\mathcal{S}_{s}}(\eta )\big)^{+}}{\mathbb{I}_{\{{\eta _{s-}}={\mathcal{S}_{s-}}(\eta )\}}}=0.\end{aligned}\]]]>
By the property of the Snell envelope, we know that ∫0T(St−(η)−ηt−)dK˜t=0 (see Lemma A.4 in [15]), i.e.
0=∫0T(St−(η)−ηt−)dK˜t=∫0T(Yt−−Lt−)dK˜t.
Therefore, we have ∫0T(Y˜t−Lt)dK˜tc=0.
The construction is now complete. Additionally, the fact that (Y˜,Z˜,V˜,K˜) is in Dμ2 results from estimations similar to those of Corollary 1.
Now let (Y,Z,V) and (Y′,Z′,V′)∈Lμ2 be such that
Ψ(Y,Z,V)=(Y˜,Z˜,V˜),Ψ(Y′,Z′,V′)=(Y˜′,Z˜′,V˜′).
Applying Itô’s formula, for γ>00$]]> one has
E[eγt+μAt(Y˜t−Y˜t′)2]+γE∫tTeγs+μAs|Y˜s−Y˜s′|2ds+μE∫tTeγs+μAs|Y˜s−Y˜s′|2dAs+E∫tTeγs+μAs|Z˜s−Z˜s′|2ds+E∫tTeγs+μAs‖V˜s−V˜s′‖λ2ds≤2E∫tTeγs+μAs(Y˜s−Y˜s′)(f(s,Ys,Zs,Vs)−f(s,Ys′,Zs′,Vs′))ds+2E∫tTeγs+μAs(Y˜s−Y˜s′)(g(s,Ys)−g(s,Ys′))dAs+2E∫tTeγs+μAs(Y˜s−−Y˜s−′)(dK˜s−dK˜s′).
First, let us show that
E∫tTeγs+μAs(Y˜s−−Y˜s−′)(dK˜s−dK˜s′)≤0.
Since Y˜ and Y˜′ belong to Sμ2 and their jumps are nonpositive, the sets
δ(ω):={t∈[0,T],ΔtY˜≠0}andδ′(ω):={t∈[0,T],ΔtY˜′≠0}
are at most countable. Using the Skorokhod condition, it yields
∫tTeγs+μAs(Y˜s−−Y˜s−′)(dK˜sc−dK˜s′c)=−∫tTeγs+μAs(Y˜s−Ls)dK˜s′c+∫tTeγs+μAs(Ls−Y˜s′)dK˜sc≤0.
On the other hand,
∫tTeγs+μAs(Y˜s−−Y˜s−′)(dK˜sd−dK˜s′d)=∫tTeγs+μAs(Y˜s−−Y˜s−′)dK˜sd−∫tTeγs+μAs(Y˜s−−Y˜s−′)dK˜s′d.
Let us deal with the second part of the right-hand side of (6). We have
∫tTeγs+μAs(Y˜s−−Y˜s−′)dK˜s′d=∑t<s≤Teγs+μAsΔsK˜dΔsK˜′d+∫tTeγs+μAs(Y˜s−Y˜s′)dK˜s′d−∑t<s≤Teγs+μAs(ΔsY˜′)2
and
∫tTeγs+μAs(Y˜s−Y˜s′)dK˜s′d=∫tTeγs+μAs(Y˜s−Ls−)dK˜s′d+∑t<s≤Teγs+μAs(ΔsY˜′)2.
Afterwards, we have
∫tTeγs+μAs(Y˜s−Ls−)dK˜s′d=−∑t<s≤Teγs+μAsΔsK˜dΔsK˜′d+∫tTeγs+μAs(Y˜s−Ls−)𝟙{ΔsY˜=0}dK˜s′d≥−∑t<s≤Teγs+μAsΔsK˜dΔsK˜′d,
since ∫tTeγs+μAs(Y˜s−Ls−)𝟙{ΔsY˜=0}dK˜s′d≥0. In conclusion,
∫tTeγs+μAs(Y˜s−−Y˜s−′)dK˜s′d≥0.
In the same way, we can prove that
∫tTeγs+μAs(Y˜s−−Y˜s−′)dK˜sd≤0.
Now coming back to (5), one has
γE∫0Teγs+μAs|Y˜s−Y˜s′|2ds+μE∫0Teγs+μAs|Y˜s−Y˜s′|2dAs+E∫0Teγs+μAs[|Z˜s−Z˜s′|2+‖V˜s−V˜s′‖λ2]ds≤εκ′E∫0Teγs+μAs|Y˜s−Y˜s′|2ds+3κ′εE∫0Teγs+μAs[|Ys−Ys′|2+|Zs−Zs′|2+‖Vs−Vs′‖λ2]ds+εκ′E∫0Teγs+μAs|Y˜s−Y˜s′|2dAs+κ′εE∫0Teγs+μAs|Ys−Ys′|2dAs.
Hence
(γ−εκ′)E∫0Teγs+μAs|Y˜s−Y˜s′|2ds+(μ−εκ′)E∫0Teγs+μAs|Y˜s−Y˜s′|2dAs+E∫0Teγs+μAs[|Z˜s−Z˜s′|2+‖V˜s−V˜s′‖λ2]ds≤3κ′ε[E∫0Teγs+μAs[(Ys−Ys′)2+|Zs−Zs′|2+‖Vs−Vs′‖λ2]ds+E∫0Teγs+μAs(Ys−Ys′)2dAs].
Now, let γ,μ>11$]]> and ε be such that 3κ′<ε<κ′−1(max{γ,μ}−1). Then Ψ is a contraction mapping on Lμ2. Henceforth, there exists a triple of processes (Y,Z,V) that is a fixed point of Ψ which, with K, is the unique solution of the reflected generalized BSDE (3). □
The next proposition gives the second step. This result is the key of our proof. We assume that the coefficient f does not depend on the variables (z,v).
Suppose that (H1)–(H3) hold. Then for any(Z,V)∈Hμ2×Lμ,λ2, the reflected generalized BSDEYt=ξ+∫tTf(s,Ys,Zs,Vs)ds+∫tTg(s,Ys)dAs+(KT−Kt)−∫tTZsdWs−∫tT∫UVs(e)N˜(ds,de),0≤t≤T,Yt≥Ltand∫0T(Yt−−Lt−)dKt=0,0≤t≤T,has a unique solution.
Let f(t,y,Zt,Vt)=h(t,y). Considering Remark 1-(2), we shall assume that α≡0 in the remaining part of this section. Then some assumptions in (H2) on the function h will be change as follows:
(y−y′)(h(t,y)−h(t,y′))≤0,
|h(t,y)|≤φ˜t+κ|y| and |g(t,y)|≤ψt+κ|y| such that
φ˜t=φt+κ|Zs|+κ‖Vs‖λandE∫0TeμAt|φ˜t|2dt+E∫0TeμAt|ψt|2dAt<+∞,
y↦(h(t,y),g(t,y)) is continuous dP×dt a.e.
We split the proof in two parts.
Part 1. We suppose that
|ξ|+sup0≤t≤T|φt˜|+sup0≤t≤T|ψt|+sup0≤t≤T|Lt+|≤M.
What we would like to do is to construct a sequences of Lipschitz (globally in y, uniformly w.r.t. (ω,s)) functions hn and gn which approximate h and g and which are monotone. However, we only manage to construct a sequence for which each hn (resp. gn) is monotone in a given ball (the radius depends on n). As we will see later in the proof, this “local” monotonicity is sufficient to obtain the result. We shall use an approximate identity.
Let a function ρ:R→R+ be in C∞ and with a compact support in the unit ball such that ∫ρ(u)du=1. For any n≥1, we put ρn(u)=nρ(nu).
Moreover, let θq:R→[0,1] be in C∞ such that θq(y)=1 if |y|≤q and θq(y)=0 if |y|≥q+1, where the value of q(n) will be fixed later. For n≥1, we set:
ξ˜=𝟙{eμAT≤n}ξ,hn(t,y)=𝟙{eμAt≤n}(ρn∗θq(n)+2h(t,.))(y),gn(t,y)=𝟙{eμAt≤n}(ρn∗θq(n)+2g(t,.))(y),Lt˜=𝟙{eμAT≤n}Lt.
Clearly,
hn(t,y)→h(t,y) and gn(t,y)→g(t,y) as n→∞,
ξ˜ and Lt˜ satisfies respectively (H1) and (H3),
hn satisfies (H2)-(i)′-(ii)-(iv)′-(v)′ and gn satisfies (H2)-(iii)-(iv)′-(v)′ and also they are Lipschitz in y uniformly w.r.t. (t,ω). In fact, taking into consideration that ρ is with a compact support in the unit ball, we have
|∇hn(t,y)|≤Cn′and|∇gn(t,y)|≤Cn′,
where Cn′=n(M+κq(n)+3κ).
Then, for any n≥1, from Proposition 3 there exists a unique process (Yn,Zn,Vn,Kn) satisfying (3) and
Ytn=ξ˜+∫tThn(s,Ysn)ds+∫tTgn(s,Ysn)dAs+(KTn−Ktn)−∫tTZsndWs−∫tT∫UVsn(e)N˜(ds,de),0≤t≤T,Ytn≥Lt˜and∫0T(Yt−n−L˜t−)dKtn=0,0≤t≤T.
Under assumptions (H1)–(H3) and with a computations similar to those in Corollary 1, there exists a constant C independ of n, such that
supnEsup0≤t≤TeμAt|Ytn|2+∫0TeμAs|Ysn|2dAs+∫0TeμAs[|Zsn|2+‖Vsn‖λ2]ds+|KTn|2≤C.
Now the rest of this part is based on the following lemmas.
Under (H1)–(H3), (8), (9) and withγ,μ>11$]]>, we have|Ytn|2≤K(n),s.t.K(n)=c0+c1n+c2n2andq(n)=[K12], where[r]is the integer part of r.
This justifies the choice of the integer q(n) above.
By virtue of Itô’s formula with γ,μ>11$]]>, we have
|Ytn|2+γ∫tTeγ(s−t)+μ(As−At)|Ysn|2ds+μ∫tTeγ(s−t)+μ(As−At)|Ysn|2dAs+∫tTeγ(s−t)+μ(As−At)|Zsn|2ds+∫tT∫Ueγ(s−t)+μ(As−At)|Vsn|2N(ds,de)=eγ(T−t)+μ(AT−At)|ξ˜|2+2∫tTeγ(s−t)+μ(As−At)Ysnhn(s,Ysn)ds+2∫tTeγ(s−t)+μ(As−At)Ysngn(s,Ysn)dAs+2∫tTeγ(s−t)+μ(As−At)YsnZsndWs+2∫tT∫Ueγ(s−t)+μ(As−At)Ys−nVsn(e)N˜(ds,de)+∫tTeγ(s−t)+μ(As−At)Ys−ndKsn.
Taking conditional expectation E(.|Ft)≜EFt(.), we obtain
|Ytn|2+γEFt∫tTeγ(s−t)+μ(As−At)|Ysn|2ds+μEFt∫tTeγ(s−t)+μ(As−At)|Ysn|2dAs+EFt∫tTeγ(s−t)+μ(As−At)[|Zsn|2+‖Vsn‖λ2]ds≤EFteγ(T−t)+μ(AT−At)|ξ˜|2+2EFt∫tTeγ(s−t)+μ(As−At)Ysnhn(s,Ysn)ds+EFt∫tTeγ(s−t)+μ(As−At)Ysngn(s,Ysn)dAs+EFt∫tTeγ(s−t)+μ(As−At)L˜s−dKsn.
From assumption (H2), for all y we have
2yhn(s,y)≤𝟙{eμAs≤n}((γ−1)y2+1γ−1φs˜2),
and
2ygn(s,y)≤𝟙{eμAs≤n}((μ−1)y2+1μ−1ψs2).
Coming back to (13), using the above inequalities and taking into account the assumption (8), one get for ρ>00$]]>,
|Ytn|2+EFt∫tTeγ(s−t)+μ(As−At)|Ysn|2ds+EFt∫tTeγ(s−t)+μ(As−At)|Ysn|2dAs+EFt∫tTeγ(s−t)+μ(As−At)[|Zsn|2+‖Vsn‖λ2]ds≤eγTM2.n+eγT.T.M2.nγ−1+eγT.M2.nμ(μ−1)+ρ.eγT.n2.M2+1ρEFt|KTn−Ktn|2.
Again by (H2), (8), Hölder’s inequality and isometry property (see, e.g., Theorem 2.3.3, p. 23 in [4]), we end up with
EFt|KTn−Ktn|2≤6{|Ytn|2+EFt|ξ˜|2+EFt(∫tT|hn(s,Ysn)|ds)2+EFt(∫tT|gn(s,Ysn)|dAs)2+EFt|∫tTZsndWs|2+EFt|∫tT∫UVsn(e)N˜(ds,de)|2}≤6M2+12T2M2+12μ2M2.n+6|Ytn|2+12κ2TEFt∫tT|Ysn|2ds+12κ2μEFt∫tTeμAs|Ysn|2dAs+6EFt∫tT[|Zsn|2+‖Vsn‖λ2]ds.
Plugging (15) in (14), we get
(1−6ρ)|Ytn|2+(1−12κ2Tρ)EFt∫tTeγ(s−t)+μ(As−At)|Ysn|2ds+(1−12κ2μρ)EFt∫tTeγ(s−t)+μ(As−At)|Ysn|2dAs+(1−6ρ)EFt∫tTeγ(s−t)+μ(As−At)[|Zsn|2+‖Vsn‖λ2]ds≤(eγTM2.n+eγT.T.M2.nγ−1+eγT.M2.nμ(μ−1)+ρ.eγT.n2.M2)+1ρ(6M2+12T2M2+12μ2M2.n).
Choosing ρ such that
ρ>max{6;12κ2T;12κ2μ},\max \bigg\{6;12{\kappa ^{2}}T;\frac{12{\kappa ^{2}}}{\mu }\bigg\},\]]]>
we obtain
|Ytn|2≤c0+c1n+c2n2.
□
The processes(Yn,Zn,Vn,Kn)converge inDμ.
Proof of Lemma2. First, mention that hn and gn are not necessary monotone on the entire space considered first, but they are monotone in the ball with the center 0 and radius q(n)+1. In fact, we have for any |y|,|y′|≤q(n)+1,
(y−y′)(hn(t,y)−hn(t,y′))=∫ρn(u).(y−y′)(h(t,y−u)−h(t,y′−u))du≤0,
and
(y−y′)(gn(t,y)−gn(t,y′))=∫ρn(u).(y−y′)(g(t,y−u)−g(t,y′−u))du≤0.
Let m≥n. Using Itô’s formula and taking expectation, we end up with
E[eγt+μAt|Ytm−Ytn|2]+γE∫tTeγs+μAs|Ysm−Ysn|2ds+μE∫tTeγs+μAs|Ysm−Ysn|2dAs+E∫tTeγs+μAs[|Zsm−Zsn|2+‖Vsm−Vsn‖λ2]ds≤2E∫tTeγs+μAs(Ysm−Ysn)(hm(s,Ysm)−hn(s,Ysn))ds+2E∫tTeγs+μAs(Ysm−Ysn)(gm(s,Ysm)−gn(s,Ysn))dAs+2E∫tTeγs+μAs(Ys−m−Ys−n)(dKsm−dKsn).
We cannot use the priori estimates because the functions hm, hn, gm and gn are not globally monotone. Nevertheless, hm and gm are monotone on the ball with radius a=q(m)+1. Since |Ysm|≤q(m)+1 and |Ysn|≤q(n)+1≤q(m)+1, in view of (12), Ym and Yn belong to this ball. As a result,
(Ysm−Ysn)(hm(s,Ysm)−hn(s,Ysn))≤2asup|y|≤a|hm(s,y)−hn(s,y)|,
and
(Ysm−Ysn)(gm(s,Ysm)−gn(s,Ysn))≤2asup|y|≤a|gm(s,y)−gn(s,y)|.
On the other hand, we have
∫0Teγs+μAs(Ys−m−Ys−n)(dKsm−dKsn)≤0.
This implies that
E∫0Teγs+μAs[|Zsm−Zsn|2+‖Vsm−Vsn‖λ2]ds≤4aE∫0Teγs+μAssup|y|≤a|hm(s,y)−hn(s,y)|ds+4aE∫0Teγs+μAssup|y|≤a|gm(s,y)−gn(s,y)|dAs.
Since y→h(t,y) and y→g(t,y) are continuous, hn(t,.) converges towards h(t,.) and gn(t,.) converges towards g(t,.) uniformly on the compact λ⊗P a.s. Moreover, it follows from assumption (H2)-(iv)′ and the dominated convergence theorem that (Zn,Vn) is a Cauchy sequence.
Using the Burkholder–Davis–Gundy inequality, we can prove that
E[sup0≤t≤TeμAt|Ytm−Ytn|2]→m,n→∞0.
This implies that Yn is a Cauchy sequence in Sμ2. Then (Yn,Zn,Vn) converges in Lμ2, i.e. there exists a process (Y,Z,V) such that Yn→Y in Sμ2∩Hμ,A2, Zn→Z in Hμ2 and Vn→V in Lμ,λ2.
Finally, for any n≥0, we have
Ktn=Y0n−Ytn−∫0Thn(s,Ysn)ds−∫0Tgn(s,Ysn)dAs+∫0TZsndWs+∫0T∫UVsnN˜(ds,de).
Using the same argument, we get also
E[sup0≤t≤TeμAt|Ktm−Ktn|2]→m,n→∞0.
Then (Yn,Zn,Vn,Kn) is a Cauchy sequence in Dμ. □
Now we will end this part by demonstrating that the limiting process (Y,Z,V,K) of (Yn,Zn,Vn,Kn) in Dμ is a solution of the reflected generalized BSDE (7). Actually, passing to the limit in the reflected generalized BSDE driven by ξ˜, hn, gn and L˜, we have that Ytn converge to Yt in Sμ2∩Hμ,A2 and we have also the following convergences in L2 which are resulted by the martingales representation:
∫tTZsndWs→∫tTZsdWs,∫tT∫UVsn(e)N˜(ds,de)→∫tT∫UVs(e)N˜(ds,de).
On the other hand,
E[sup0≤t≤T|∫tT(hn(s,Ysn)−h(s,Ys))ds|2]→0,
and using Remark 1-(1), we get
E[sup0≤t≤T|∫tT(gn(s,Ysn)−g(s,Ys))dAs|2]→0.
Also, we have the convergence
Kt=Y0−Yt−∫0th(s,Ys)ds−∫0tg(s,Ys)dAs+∫0tZsdWs+∫0t∫UVs(e)N˜(ds,de).
Note that (Kt)t is an increasing process and E|KT|2<+∞.
Part 2 (General case). For any p≥1, we consider
ξp=(p∧|ξ|)|ξ|ξ,ifξ≠0,0,ifξ=0,hp(t,y)=h(t,y)−h(t,0)+(p∧|h(t,0)|)|h(t,0)|h(t,0),ifh(t,0)≠0,h(t,y),ifh(t,0)=0,gp(t,y)=g(t,y)−g(t,0)+(p∧|g(t,0)|)|g(t,0)|g(t,0),ifg(t,0)≠0,g(t,y),ifg(t,0)=0,Ltp=(p∧supt(Lt)+)supt(Lt)+Lt,ifsupt(Lt)+≠00,ifsupt(Lt)+=0.
It is easy to see that, when p→∞,
E[eμAT|ξp−ξ|2+∫0TeμAt|hp(t,0)−h(t,0)|2dt+∫0TeμAt|gp(t,0)−g(t,0)|2dAt]→0.
Obviously, (ξp,hp,gp,Lp) satisfies the hypothesis of the previous step. Then the reflected generalized BSDE associated to the parameters (ξp,hp,gp,Lp) has a unique solution (Yp,Zp,Vp,Kp) in Dμ.
Now we are going to show that the sequence (Yp,Zp,Vp,Kp) is a Cauchy sequence in Dμ. Let q≥p, using Itô’s formula, the definition of ξp, hp, gp, Lp and the same arguments in Proposition 1 with the Burkholder–Davis–Gundy inequality, we end up with
E[sup0≤t≤TeμAt|Ytq−Ytp|2+∫0TeμAs|Ysq−Ysp|2dAs+∫0TeμAs[|Zsq−Zsp|2+‖Vsq−Vsp‖λ2]ds]≤CE[eμAT|ξq−ξp|2+∫0TeμAs|hq(s,0)−hp(s,0)|2ds+∫0TeμAs|gq(s,0)−gp(s,0)|2dAs+∫0TeμAs[(Ls−q−Ys−p)dKsq−(Ys−q−Ls−p)dKsp]].
Remark that, since q≥p, we have Lq≥Lp, 0≤t≤T. Then
(Ls−q−Ys−p)dKsq≤(Ls−q−Ls−p)dKsq.
Since Lt−Ltn↘0 and (Lt−Ltn) is a rcll process, by the generalized Dini’s theorem (see [3, p. 202]), the convergence holds uniformly in [0,T], i.e.
limn→∞E[sup0≤s≤T|eμAs(Ls−Lsn)|2]→0asn→∞.
Then using Remark 3, we get
E∫0TeμAs(Ls−q−Ls−p)dKsq≤E[sup0≤s≤T|eμAs(Lsq−Lsp)|2]12.(E|KTq|2)12→0asp,q→∞.
The right-hand side of (17) tends to 0 as p,q→∞.
Finally, for any p≥1, we have
Ktp=Y0p−Ytp−∫0Thp(s,Ysp)ds−∫0Tgp(s,Ysp)dAs+∫0TZspdWs+∫0T∫UVspN˜(ds,de).
We also get
E[sup0≤t≤TeμAt|Ktq−Ktp|2]→q,p→∞0.
Thus the sequence (Yp,Zp,Vp,Kp) is a Cauchy sequence in Dμ. Then it converges towards a progressively measurable process (Y,Z,V,K). It remains to verify that the limiting process solves the reflected generalized BSDE (7). Hence, by the same argument as in Part 1, the process (Y,Z,V,K) is a solution of the reflected generalized BSDE (7) and the proof is complete. □
With the help of Proposition 4, we can now construct a solution (Y,Z,V,K) to the reflected generalized BSDE (3). We claim the following result.
Suppose that the assumptions (H1)–(H3) hold. Then the reflected generalized BSDE (3) has a unique solution.
The uniqueness is already established in Proposition 2. For the existence, we will use a fixed point argument. We define the map Ψ of (Lμ2, ‖.‖γ,μ) into itself as follows: for every (Y,Z,V)∈Lμ2 we put Ψ(Y,Z,V)=(Y˜,Z˜,V˜) where (Y˜,Z˜,V˜,K˜) is the solution of the reflected generalized BSDE associated with (ξ,f(t,Y˜,Z,V),g(t,Y˜),L) which exists by Proposition 4.
Now (Y˜,Z˜,V˜)∈Lμ2 is a solution of the reflected generalized BSDE (3) if and only if it is a fixed point of Ψ. Let (Y,Z,V),(Y′,Z′,V′)∈Lμ2 be such that Ψ(Y,Z,V)=(Y˜,Z˜,V˜) and Ψ(Y′,Z′,V′)=(Y˜′,Z˜′,V˜′).
Applying Itô’s formula and taking expectation, one has
E[eγt+μAt|Y˜t−Y˜t′|2]+γE∫tTeγs+μAs|Y˜s−Y˜s′|2ds+μE∫tTeγs+μAs|Y˜s−Y˜s′|2dAsE∫tTeγs+μAs[|Z˜s−Z˜s′|2+‖V˜s−V˜s′‖λ2]ds≤2E∫tTeγs+μAs(Y˜s−Y˜s′)(f(s,Y˜s,Zs,Vs)−f(s,Y˜s′,Zs′,Vs′))ds+2E∫tTeγs+μAs(Y˜s−Y˜s′)(g(s,Y˜s)−g(s,Y˜s′))dAs+2E∫tTeγs+μAs(Y˜s−−Y˜s−′)(dK˜s−dK˜s′).
In particular, if we use assumption (H2) and Remark 1-(2), we get, by choosing γ≥1+4κ2 and μ≥1, that
E∫0Teγs+μAs(Y˜s−Y˜s′)2ds+E∫0Teγs+μAs(Y˜s−Y˜s′)2dAs+E∫0Teγs+μAs[|Z˜s−Z˜s′|2+‖V˜s−V˜s′‖λ2]ds≤12E∫0Teγs+μAs[|Zs−Zs′|2+‖Vs−Vs′‖λ2]ds≤12E[∫0Teγs+μAs[|Ys−Ys′|2+|Zs−Zs′|2+‖Vs−Vs′‖λ2]ds+∫0Teγs+μAs|Ys−Ys′|2dAs].
Then
‖(Y˜−Y˜′),(Z˜−Z˜′),(V˜−V˜′)‖γ,μ2≤12‖(Y−Y′),(Z−Z′),(V−V′)‖γ,μ2.
Then Ψ is a contraction mapping on (Lμ2, ‖.‖γ,μ). Henceforth, there exists a triple of processes (Y,Z,V) that is a fixed point of Ψ which, with K, is the unique solution of the reflected generalized BSDE (3). □
Comparison theorem
In general, we do not have a comparison result for solutions of BSDEs driven by a Brownian motion and an independent Poisson process, reflected or not (see, e.g., [1] for a counter-example). But in some specific cases, when the coefficients satisfy some properties, we have a comparison result.
We establish two results of the comparison. The first one is when the coefficient f does not depend on the variable v. But in the second one, we impose some monotonicity w.r.t. v. So assume there exists another quadruple of processes (Y′,Z′,V′,K′) being a solution for the reflected generalized BSDE with one lower rcll reflecting barrier associated with (ξ′,f′,g′,L).
Assume that:
f is independent of v;
P-a.s, for anyt≤T,f(t,Yt′,Zt′)≤f′(t,Yt′,Zt′,Vt′),g(t,Yt′)≤g′(t,Yt′)andξ≤ξ′.
ThenP-a.s., for anyt≤T,Yt≤Yt′. Additionally, iff′does not depend on v then we have alsoKt−Ks≥Kt′−Ks′, for any0≤s≤t≤T.
Using Remark 2-(2), since Y≤Y′, we obviously have P-a.s., for any s≤t,
Ktd−Ksd≥Kt′d−Ks′d.
If the barriers are not the same, as it is assumed in the previous theorem, we can still get the comparison result of Ys, but the comparison of Ks could fail.
The second result extends the comparison results in [27] to the case of reflected generalized BSDE with monotone generators. The three assumptions (H1), (H2) and (H3) hold, but (H2)-(ii) is replaced by:
(H2)-(ii)′: f is Lipschitz continuous w.r.t. z with constant κ, and for each(y,z,v,v′)∈R×Rd×(Lλ)2, there exists a predictable processK=Ky,z,v,v′:Ω×[0,T]×U→Rsuch thatf(t,y,z,v)−f(t,y,z,v′)≤∫U(v(e)−v′(e))Kty,z,v,v′(e)λ(de)withP⊗m⊗λ-a.e. for any(y,z,v,v′)
−1≤Kty,z,v,v′(e),
|Kty,z,v,v′(e)|≤w(e)wherew∈Lλ.
Now, we are able to establish our comparison theorem.
Let(Y,Z,V,K)and(Y′,Z′,V′,K′)be solutions of the reflected generalized BSDE with one rcll reflecting barrier associated, respectively, with(ξ,f,g,L)and(ξ′,f′,g′,L)which satisfy all the assumptions (H1)–(H3). Assume that:
ξ≤ξ′,P-a.s.;
P-a.s, for anyt≤T,f(t,Yt,Zt,Vt)≤f′(t,Yt,Zt,Vt)andg(t,Yt)≤g′(t,Yt).
ThenP-a.s., for anyt≤T,Yt≤Yt′. Additionally, if f andf′do not depend on v, then we have alsoKt−Ks≥Kt′−Ks′, for any0≤s≤t≤T.
Applications to the obstacle problem for integral-partial differential equations with Neumann boundary condition
With the help of BSDEs, the Feynman–Kac formula provides a probabilistic interpretation for semilinear second-order PDEs of elliptic or parabolic types, which has been generalized to systems of quasilinear second-order PDEs by Peng [22], Pardoux and Tang [20], see also Darling and Pardoux [2] and references therein. The case of PDE with nonlinear Neumann boundary conditions have been first treated by Pardoux and Zhang [17] and extended to several cases, see, e.g., [7, 24, 25]. Through the case of BSDEs with jumps these results have been generalized to treat a class of second-order integral-partial differential equations (IPDEs), see, e.g., [1].
The main result of this section is to prove that the solution of the reflected generalized BSDE (3) provides a probabilistic formula for a viscosity solution for an obstacle problem of a class of second-order integral-partial differential equations (IPDEs) of parabolic type with nonlinear Neumann boundary condition.
A class of reflected diffusion process
First of all let us recall some notions. Let G be an open connected bounded domain of Rl(l≥1). We suppose that G is a smooth domain, which is such that for a function Φ∈Cb2(R), G and its boundary ∂G are characterized by G={Φ>0}0\}$]]>, ∂G={Φ=0} and for any x∈∂G, ∇Φ(x) is the unit normal vector pointing toward the interior of G. In addition, the interior sphere condition holds (see [21, p. 551]), i.e. there exists m>00$]]> such that for any x∈∂G, x′∈G¯,
|x′−x|2+m⟨∇Φ(x),x′−x⟩≥0.
Now from [14], we know that for every (t,x)∈R+×G¯ there exists a unique pair of progressively measurable process (Xst,x,Ast,x)s≥0 being a solution to the following reflected stochastic differential equation (reflected SDE) with jumps:
Xst,x=x+∫tt∨sb(Xrt,x)dr+∫tt∨sσ(Xrt,x)dWr+∫tt∨s∫Uc(Xr−t,x,e)N˜(dr,de)+∫tt∨s∇Φ(Xrt,x)dArt,x,0≤s≤T,Ast,x=∫tt∨s𝟙{Xrt,x∈∂G}dArt,x,
where b:Rl→Rl and σ:Rl→Rl×l satisfy, for κ>00$]]> and for any (x,x′)∈Rl+l,
(i)|b(x)−b(x′)|+|σ(x)−σ(x′)|≤κ|x−x′|,(ii)|b(x)|+|σ(x)|≤κ(1+|x|).
Moreover c:Rl×U→R is a measurable function which satisfies, for any e∈U and x,x′∈Rl, the following conditions:
(i)|c(x,e)|≤κ(1∧|e|),(ii)|c(x,e)−c(x′,e)|≤κ|x−x′|(1∧|e|).
Under assumptions (18), (20) and (21) we state some properties of the processes (Xst,x,Ast,x)s≥0 which can be found in [14] and [10].
For eachT≥0, there exists two constantsCTandCT′such that for allt<t′≤Tandx,x′∈G¯,E[supt′≤s≤T|Xst,x−Xst′,x′|4]≤CT(|x−x′|4+|t−t′|2),E[supt′≤s≤T|Ast,x−Ast′,x′|4]≤CT′(|x−x′|4+|t−t′|2).Furthermore for all0≤t≤s≤r, we haveXrt,x=Xrs,Xst,x.
Characterization (22) implies the Markov property of the process (Xst,x)s≥0 that allows the relationship with integral-partial differential equations to be established.
Viscosity solution for the obstacle IPDE with Neumann boundary condition
For all (t,x)∈[0,T]×G¯, let (Xst,x,Ast,x)s≥0 denote the solution of the reflected SDE (19). Let us set
ξt,x:=H(XTt,x),Ls(ω):=ℓ(s,Xst,x),f(s,ω,y,z,v):=f(s,Xst,x,y,z,∫Uv(e)γ(Xst,x,e)λ(de)),g(s,ω,y):=g(s,Xst,x,y),
where the functions f:[0,T]×G¯×R×Rd×R→R and g:[0,T]×G¯×R→R, and H:G¯→R and ℓ:[0,T]×G¯→R are continuous and satisfy, for some constants, α,β∈R and κ,C,p>00$]]>:
(i)|f(t,x,0,0,0)|+|g(t,x,0)|+|H(x)|+|ℓ(t,x)|≤C(1+|x|p),(ii)(y−y′)(f(t,x,y,z)−f(t,x,y′,z))≤α|y−y′|2,(iii)|f(t,x,y,z,v)−f(t,x,y,z′,v′)|≤κ(|z−z′|+‖v−v′‖λ),(iv)the mappingr→f(t,x,y,z,r)is nondecreasing,(v)(y−y′)(g(t,x,y)−g(t,x,y′))≤β|y−y′|2,(vi)ℓ∈C1,2such thatℓ(T,x)≤H(x),∀(t,x)∈[0,T]×G¯.
Moreover we suppose that the function γ:Rl×U→R satisfies, for any e∈U and x,x′∈G, the conditions
(i)γ(x,e)≤κ(1∧|e|),(ii)|γ(x,e)−γ(x′,e)|≤κ|x−x′|(1∧|e|).
It follows from Theorem 1 that, for all (t,x)∈[0,T]×G¯, there exists a unique quadruple (Yst,x,Zst,x,Vst,x,Kst,x)t≤s≤T being a solution of the following reflected generalized BSDE:
(i)E[supt≤s≤T|Yst,x|2+∫tT|Yst,x|2dAst,x+∫tT(|Zst,x|2+‖Vst,x‖λ2)ds]<∞,(ii)Yst,x=H(XTt,x)+∫sTf(r,Xrt,x,Yrt,x,Zrt,x,Vrt,x)dr+∫sTg(r,Xrt,x,Yrt,x)dAr+KTt,x−Kst,x−∫sTZrt,xdWr−∫sT∫UVrt,x(e)N˜(dr,de),t≤s≤T,(iii)Yst,x≥ℓ(s,Xst,x),t≤s≤T,(iv)∫tT(Ys−t,x−ℓ(s,Xs−t,x))dKst,x=0,P-a.s.
The process Yst,x is Fst-adapted and (Zst,x,Vst,x,Kst,x) are Fs-predictable where
Fst=σ(Ws−Wt,N(]t,s],Λ),t≤s≤r,Λ∈U)∨N.
Now, we consider the following related obstacle problem for a parabolic integral-partial differential equation with nonlinear Neumann boundary conditions
(u(t,x)−ℓ(t,x))∧(−∂u∂t(t,x)−Lu(t,x)−f(t,x,u(t,x),(∇xuσ)(t,x),Bu(t,x)))=0,∀(t,x)∈[0,T]×G,u(T,x)=H(x),∀x∈G,∂u∂n(t,x)+g(t,x,u(t,x))=0,∀x∈∂G,
where L is the second-order integral-differential operator
L=R+S
with
Rϕ=12Tr[σσT(x)]Dx2ϕ(t,x)+⟨b(x),∇xϕ(t,x)⟩,Sϕ=∫U(ϕ(t,x+c(x,e))−ϕ(t,x)−⟨∇xϕ(t,x),c(x,e)⟩)λ(de),
and B is an integral operator defined as
Bϕ=∫U(ϕ(t,x+c(x,e))−ϕ(t,x))γ(x,e)λ(de),
and for every x∈∂G,
∂ϕ∂n=⟨∇xϕ,∇Φ(x)⟩.
Now, according to Definition 3.1, Remark 3.2 and Lemma 3.3 in [1], we give a definition of the viscosity solution of (27).
Let u be a function which belongs to C([0,T]×G¯,R). Then:
It is called a subsolution of (27), if u(T,x)≤H(x), ∀x∈G¯, and for any φ∈C1,2([0,T]×G¯) such that whenever (t,x)∈[0,T]×G¯ is a local maximum of u−φ, we have, suppressing dependence on (t,x),
(u−ℓ)∧[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]≤0,x∈G,[(u−ℓ)∧[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]]∧[−∂φ∂n−g(t,x,u)]≤0,x∈∂G.
In other words, if u(t,x)>ℓ(t,x)\ell (t,x)$]]> then
[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]≤0,x∈G,[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]∧[−∂φ∂n−g(t,x,u)]≤0,x∈∂G.
It is called a supersolution of (27), if u(T,x)≥H(x), ∀x∈G¯, and for any φ∈C1,2([0,T]×G¯) such that whenever (t,x)∈[0,T]×G¯ is a local minimum of u−φ, we have, suppressing dependence on (t,x),
(u−ℓ)∧[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]≥0,x∈G,[(u−ℓ)∧[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]]∨[−∂φ∂n−g(t,x,u)]≥0,x∈∂G.
In other words, if u(t,x)≥ℓ(t,x) then
[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]≥0,x∈G,[−φt−Lφ−f(t,x,u,(∇φσ),Bφ)]∨[−∂φ∂n−g(t,x,u)]≥0,x∈∂G.
u∈C([0,T]×G¯) is said to be a viscosity solution of (27), if it is both sub- and supersolution.
Now, we denote
u(t,x)=Ytt,x,
Obviously, this function is deterministic. By the uniqueness of the solution of (26), it is not hard to see that
Yst,x=Yss,Xst,x=u(s,Xst,x),∀t≤s<T.
Next we will indicate some basic properties of this function.
u∈C([0,T]×G¯,R).
First, we define for all (t,x) the solution Yst,x for all s∈[0,T] by choosing Yst,x=Ytt,x for 0≤s≤t. Note that, for each sequence (tn,xn) which converges to (t,x), the following converges as n→∞:
E[eμkT|H(XTt,x)−H(XTtn,xn)|2]⟶n→∞0,E∫0Teμkr|f(r,Xrt,x,Yrt,x,Zrt,x,Vrt,x)−f(r,Xrtn,xn,Yrt,x,Zrt,x,Vrt,x)|2dr⟶n→∞0,E∫0Teμkr|g(r,Xrt,x,Yrt,x)−g(r,Xrtn,xn,Yrt,x)|2dArtn,xn⟶n→∞0,E∫0Teμkr|g(r,Xrt,x,Yrt,x)|2(dArt,x−dArtn,xn)⟶n→∞0,E∫0Teμkr(ℓ(r,Xst,x)−ℓ(r,Xstn,xn))2ΔKr⟶n→∞0,
with k≜|A¯|+Atn,xn where A¯=At,x−Atn,xn and |A¯| is the total variation of A¯ and ΔK:=Kt,x−Ktn,xn. Those convergences follow from the continuity assumptions of f, g, H and ℓ and Proposition 5. As in the proof of Proposition 1, we can derive the desired result. □
We now prove that our reflected generalized BSDE provides a viscosity solution of (27).
The function u defined in (28) is a viscosity solution of (27).
First let us show that u is a viscosity subsolution of (27). A similar argument would show that is a viscosity supersolution of (27). Let φ∈C1,2([0,T]×G¯) and (t0,x0)∈[0,T]×G¯ such that φ(t0,x0)=u(t0,x0) and φ(t,x)≥u(t,x) for all (t,x)∈[0,T]×G¯.
Step 1. Suppose that u(t0,x0)>ℓ(t0,x0)\ell ({t_{0}},{x_{0}})$]]> and x0∈G, and that
−φt(t0,x0)−Lφ(t0,x0)−f(t0,x0,u(t0,x0),(∇φσ)(t0,x0),Bφ(t0,x0))>0,0,\]]]>
and we will find a contradiction.
It follows from the continuity of f, g, b, σ, c and φ that there exist ε>00$]]> and ηε>00$]]> such that for all (t,x), t0≤t≤t0+ηε and {x:|x−x0|≤ηε}⊂G, we have u(t,x)≥ℓ(t,x)+ε and
−φt(t,x)−Lφ(t,x)−f(t,x,u(t,x),(∇φσ)(t,x),Bφ(t,x))≥ε.
Define
τ=inf{s≥t0:|Xst0,x0−x0|>ηε}∧(t0+ηε).{\eta _{\varepsilon }}\big\}\wedge ({t_{0}}+{\eta _{\varepsilon }}).\]]]>
Note that for all s∈[t0,τ], we have
u(s,Xst0,x0)≥ℓ(s,Xst0,x0)+ε.
Consequently, the process (Kt0,x0) is constant in [t0,τ] and, for all t0≤s≤τ, we have
Yst0,x0=Yτt0,x0+∫sτf(r,Xrt0,x0,Yrt0,x0,Zrt0,x0,Vrt0,x0)dr−∫sτZrt0,x0dWr−∫sτ∫UVrt0,x0(e)N˜(dr,de).
On the other hand, applying Itô’s formula to φ(s,Xst0,x0) yields that
φ(τ,Xτt0,x0)=φ(s,Xst0,x0)+∫sτ∂φ∂r(r,Xrt0,x0)dr+∫sτ∇φ(r,Xrt0,x0)dXrt0,x0+12∫sτD2φ(r,Xrt0,x0)(σσT)(Xrt0,x0)dr+∫sτ∫U[φ(r,Xr−t0,x0+c(Xr−t0,x0,e))−φ(r,Xr−t0,x0)−∇φ(r,Xr−t0,x0)c(Xr−t0,x0,e)]N(dr,de)=φ(s,Xst0,x0)+∫sτ[∂φ∂r(r,Xrt0,x0)+∇φ(r,Xrt0,x0)b(Xrt0,x0)+12D2φ(r,Xrt0,x0)(σσT)(Xrt0,x0)]dr+∫sτ∇φ(r,Xrt0,x0)σ(Xrt0,x0)dWr+∫sτ∫U[φ(r,Xr−t0,x0+c(Xr−t0,x0,e))−φ(r,Xr−t0,x0)]N˜(dr,de)+∫sτ∫U[φ(r,Xr−t0,x0+c(Xr−t0,x0,e))−φ(r,Xr−t0,x0)−∇φ(r,Xr−t0,x0)c(Xr−t0,x0,e)]λ(de)dr.
Then
φ(s,Xst0,x0)=φ(τ,Xτt0,x0)−∫sτ[∂φ∂r+Lφ](r,Xrt0,x0)dr−∫sτ∇φ(r,Xrt0,x0)σ(Xrt0,x0)dWr−∫sτ∫U[φ(r,Xr−t0,x0+c(Xr−t0,x0,e))−φ(r,Xr−t0,x0)]N˜(dr,de),t0≤s≤τ.
Now, by assumption (29), we have
−[∂φ∂s+Lφ](s,Xst0,x0)−f(s,Xst0,x0,φ(s,Xst0,x0),(∇φσ)(s,Xst0,x0),Bφ(s,Xst0,x0))≥ε.
Also,
φ(τ,Xτt0,x0)≥u(τ,Xτt0,x0)=Yτt0,x0.
We deduce with the help of Theorem 3 that
φ(t0,x0)>φ(t0,Xt0t0,x0)−ε(τ−t0)≥u(t0,x0)\varphi \big({t_{0}},{X_{{t_{0}}}^{{t_{0}},{x_{0}}}}\big)-\varepsilon (\tau -{t_{0}})\ge u({t_{0}},{x_{0}})\]]]>
which contradicts our assumption.
Step 2. If we continue to assume that u(t0,x0)>ℓ(t0,x0)\ell ({t_{0}},{x_{0}})$]]> and x0∈∂G, and that
[−φt(t0,x0)−Lφ(t0,x0)−f(t0,x0,u(t0,x0),(∇φσ)(t0,x0),Bφ(t0,x0))]∧[−∂φ∂n(t0,x0)−g(t0,x0,φ(t0,x0))]>0,0,\end{aligned}\]]]>
we will find a contradiction.
It follows from the continuity of f, g, b, σ, c and φ that there exist ε>00$]]> and ηε>00$]]> such that for all (t,x), t0≤t≤t0+ηε and {x:|x−x0|≤ηε}, we have u(t,x)≥ℓ(t,x)+ε and [−φt(t,x)−Lφ(t,x)−f(t,x,u(t,x),(∇φσ)(t,x),Bφ(t,x))]∧[−∂φ∂n(t,x)−g(t,x,φ(t,x))]≥ε. Let τ be the stopping time defined as above in (30), and let us note that for all s∈[t0,τ] we have
u(s,Xst0,x0)≥ℓ(s,Xst0,x0)+ε.
Consequently, the process (Kt0,x0) is constant in [t0,τ] and for all t0≤s≤τ, and we have
Yst0,x0=Yτt0,x0+∫sτf(r,Xrt0,x0,Yrt0,x0,Zrt0,x0,Vrt0,x0)dr+∫sτg(r,Xrt0,x0,Yrt0,x0)dArt0,x0−∫sτZrt0,x0dWr−∫sτ∫UVrt0,x0(e)N˜(dr,de).
On the other hand, applying Itô’s formula to φ(s,Xst0,x0) we end up with
φ(s,Xst0,x0)=φ(τ,Xτt0,x0)−∫sτ[∂φ∂r+Lφ](r,Xrt0,x0)dr+∫sτ∂φ∂n(r,Xrt0,x0)dArt0,x0−∫sτ∇φ(r,Xrt0,x0)σ(Xrt0,x0)dWr−∫sτ∫U[φ(r,Xr−t0,x0+c(Xr−t0,x0,e))−φ(r,Xr−t0,x0)]N˜(dr,de),t0≤s≤τ.
Now, by assumption (31), we have
(−[∂φ∂s+Lφ](s,Xst0,x0)−f(s,Xst0,x0,φ(s,Xst0,x0),(∇φσ)(s,Xst0,x0),Bφ(s,Xst0,x0)))∧[−∂φ∂n(s,Xst0,x0)−g(s,Xst0,x0,φ(s,Xst0,x0))]≥ε.
Also,
φ(τ,Xτt0,x0)≥u(τ,Xτt0,x0)=Yτt0,x0.
We deduce with the help of Theorem 3 that
φ(t0,x0)>φ(t0,Xt0t0,x0)−ε(τ−t0)≥Yt0t0,x0=u(t0,x0).\varphi \big({t_{0}},{X_{{t_{0}}}^{{t_{0}},{x_{0}}}}\big)-\varepsilon (\tau -{t_{0}})\ge {Y_{{t_{0}}}^{{t_{0}},{x_{0}}}}=u({t_{0}},{x_{0}}).\]]]>
which leads to a contradiction. □
The uniqueness of the viscosity solution will be obtained by the comparison of sub- and supersolutions of IPDE with nonlinear Neumann boundary conditions. It follows from an adaptation of standard techniques and the proof of Theorem 3.5 in [1], taking into account the continuity of the solution u and the obstacle ℓ which enables us to focus exclusively on the results on [0,T]×G, and when u>ℓ\ell $]]>.
Acknowledgement
The authors would like to thank the editor and referee for their careful reading, valuable comments, and numerous constructive suggestions, which significantly improved the paper’s quality.
ReferencesBarles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Int. J. Probab. Stoch. Process.60(1-2), 57–83 (1997). MR1436432. https://doi.org/10.1080/17442509708834099Darling, R.W., Pardoux, E.: Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab.25(3), 1135–1159 (1997). MR1457614. https://doi.org/10.1214/aop/1024404508Dellacherie, C., Meyer, P.A.: Probabilités et Potentiel, Théorie des Martingales, Chaps. V–VIII (1980). MR0566768Delong, Ł.: Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. Springer (2013). MR3089193. https://doi.org/10.1007/978-1-4471-5331-3El Karoui, N., Peng, S., Quenez, M.C.: Backward stochastic differential equations in finance. Math. Finance7(1), 1–71 (1997). MR1434407. https://doi.org/10.1111/1467-9965.00022El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.-C.: Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s. Ann. Probab.25(2), 702–737 (1997). MR1434123. https://doi.org/10.1214/aop/1024404416El Otmani, M.: Generalized bsde driven by a lévy process. J. Appl. Math. Stoch. Anal.2006, 085407 (2006). MR2253532. https://doi.org/10.1155/JAMSA/2006/85407El Otmani, M.: Reflected BSDE driven by a Lévy process. J. Theor. Probab.22(3), 601–619 (2009). MR2530105. https://doi.org/10.1007/s10959-009-0229-3Essaky, E.: Reflected backward stochastic differential equation with jumps and rcll obstacle. Bull. Sci. Math.132(8), 690–710 (2008). MR2474488. https://doi.org/10.1016/j.bulsci.2008.03.005Fujiwara, T., Kunita, H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ.25(1), 71–106 (1985). MR0777247. https://doi.org/10.1215/kjm/1250521160Hamadène, S., Ouknine, Y.: Reflected backward stochastic differential equation with jumps and random obstacle. Electron. J. Probab.8, 1–20 (2003). MR1961164. https://doi.org/10.1214/EJP.v8-124Hamadene, S., Ouknine, Y.: Reflected backward sdes with general jumps. Theory Probab. Appl.60(2), 263–280 (2016). MR3568776. https://doi.org/10.1137/S0040585X97T987648Hamedene, S., Lepeltier, J.: Zero-sum stochastic differential games and BSDEs. Syst. Control Lett.24, 259–263 (1995). MR1321134. https://doi.org/10.1016/0167-6911(94)00011-JŁaukajtys, W., Słomiński, L.: Penalization methods for reflecting stochastic differential equations with jumps. Stoch. Stoch. Rep.75(5), 275–293 (2003). MR2017780. https://doi.org/10.1080/1045112031000155687Lepeltier, J.-P., Xu, M.: Penalization method for reflected backward stochastic differential equations with one rcll barrier. Stat. Probab. Lett.75(1), 58–66 (2005). MR2185610. https://doi.org/10.1016/j.spl.2005.05.016Øksendal, B., Sulem, A.: Stochastic control of jump diffusions. In: Applied Stochastic Control of Jump Diffusions, pp. 93–155. Springer (2019). MR3931325. https://doi.org/10.1007/978-3-030-02781-0Pardoux, E.: Generalized discontinuous backward stochastic differential equations. In: Pitman Research Notes in Mathematics Series, pp. 207–219 (1997). MR1752684. https://doi.org/10.1016/S0377-0427(97)00124-6Pardoux, E., Peng, S.: Adapted solution of a backward stochastic differential equation. Syst. Control Lett.14(1), 55–61 (1990). MR1037747. https://doi.org/10.1016/0167-6911(90)90082-6Pardoux, E., Peng, S.: Backward stochastic differential equations and quasilinear parabolic partial differential equations. In: Stochastic Partial Differential Equations and Their Applications, pp. 200–217. Springer (1992)Pardoux, E., Tang, S.: Forward-backward stochastic differential equations and quasilinear parabolic PDEs. Probab. Theory Relat. Fields114(2), 123–150 (1999). MR1701517. https://doi.org/10.1007/s004409970001Pardoux, E., Zhang, S.: Generalized BSDEs and nonlinear Neumann boundary value problems. Probab. Theory Relat. Fields110(4), 535–558 (1998). MR1626963. https://doi.org/10.1007/s004400050158Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep.37(1-2), 61–74 (1991). MR1149116Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Applications of Mathematics, vol. 21. Springer (2004). 0172-4568. MR2020294Ren, Y., El Otmani, M.: Generalized reflected BSDEs driven by a Lévy process and an obstacle problem for PDIEs with a nonlinear Neumann boundary condition. J. Comput. Appl. Math.233(8), 2027–2043 (2010). MR2564037. https://doi.org/10.1016/j.cam.2009.09.037Ren, Y., Xia, N.: Generalized reflected BSDE and an obstacle problem for PDEs with a nonlinear Neumann boundary condition. Stoch. Anal. Appl.24(5), 1013–1033 (2006). MR2258915. https://doi.org/10.1080/07362990600870454Rong, S.: On solutions of backward stochastic differential equations with jumps and applications. Stoch. Process. Appl.66(2), 209–236 (1997). MR1440399. https://doi.org/10.1016/S0304-4149(96)00120-2Royer, M.: Backward stochastic differential equations with jumps and related non-linear expectations. Stoch. Process. Appl.116(10), 1358–1376 (2006). MR2260739. https://doi.org/10.1016/j.spa.2006.02.009Tang, S., Li, X.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim.32(5), 1447–1475 (1994). MR1288257. https://doi.org/10.1137/S0363012992233858