This paper deals with a new class of reflected backward stochastic differential equations driven by countable Brownian motions. The existence and uniqueness of the RBSDEs are obtained via Snell envelope and fixed point theorem.

1. Introduction

The nonlinear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1], who proved the existence and uniqueness of the solution under the Lipschitz conditions for giving the probabilistic interpretation of semilinear parabolic partial differential equations. Since then, many authors were devoted to studying the BSDEs (see, e.g., [2–8] and the references therein). At present, the theory of BSDEs becomes a powerful tool to solve practical matters. In 1994, Pardoux and Peng [9] firstly studied the backward doubly stochastic differential equations (BDSDEs in short), which are driven by two kinds of Brownian motions. Later, Boufoussi et al. [10] established the connection between a class of generalized BDSDEs and semilinear stochastic partial differential equations with a Neumann boundary condition.

Reflected backward differential equations (RBSDEs in short) were introduced by El Karoui et al. [11]. Later, many researchers discussed various kinds of RBSDEs for their deep application in mathematical finance and partial differential equations. Ren and Hu [12] proposed the RBSDEs, driven by Teugels martingales and Brownian motion, and derived the existence and uniqueness of the solution by means of the Snell envelope and the fixed point theorem when the barrier was right continuous with left limits. Ren and El Otmani [13] discussed the generalized reflected BSDEs driven by Lévy process. Recently, Ren et al. [14] studied a new class of reflected backward doubly stochastic differential equations driven by Lévy process and Brownian motion.

As in all the previous works, the equations are driven by finite Brownian motions. To the best of our knowledge, there are no papers on the reflected backward stochastic differential equations driven by countable Brownian motions. In this paper, we aim to derive the existence and uniqueness of the solution for the RBSDEs driven by countable Brownian motions.

The structure of the paper is organized as follows. In Section 2, we give some notations. Section 3 is devoted to the main result.

2. Notations

Let T be a positive constant. Throughout the paper (Ω,ℱ,ℙ) is a complete probability space equipped with the natural filtration {ℱt}t≥0 satisfying the usual conditions. {βj(t)}j=1∞ are mutual independent one-dimensional standard Brownian motions on the probability space. W(t) is a standard Brownian motion on ℝd which is independent of βj(t). Assume that
(1)ℱt=(⋁j=1∞ℱt,Tβj)⋁ℱtW⋁𝒩,
where for any process {ηt}, ℱs,tη=σ{ηr-ηs:s≤r≤t}, ℱtη=ℱ0,tη, and 𝒩 denotes the class of ℙ-null sets of ℱ.

For the convenience, let us introduce some spaces:

ℋ2={(φt)0≤t≤T: an ℱt-progressively measurable, ℝ-valued process such that E∫0T|φt|2dt<∞};

𝒮2={(ψt)0≤t≤T: an ℱt-progressively measurable, ℝd-valued continuous process such that E(sup0≤t≤T|ψt|2)<∞};

𝒜2={(Kt)0≤t≤T: an ℱt-adapted, continuous, increasing process such that K0=0,E|Kt|2<∞}.

With the previous preparations, we consider the following RBSDEs:
(2)Yt=ξ+∫tTf(s,Ys,Zs)ds+∑j=1∞∫tTgj(s,Ys,Zs)dβj(s)-∫tTZsdW(s)+KT-Kt,0≤t≤T,
where f:Ω×[0,T]×ℝ×ℝd→ℝ and gj:Ω×[0,T]×ℝ×ℝd→ℝ.

Definition 1.

A solution of (2) is a triple of ℝ×ℝd×ℝ+ value process (Yt,Zt,Kt)0≤t≤T, which satisfies (2), and

(Yt,Zt,Kt)0≤t≤T∈𝒮2×ℋ2×𝒜2;

Yt≥St;

Kt is a continuous and increasing process with K0=0 and ∫0T(Yt-St)dKt=0.

In order to get the solution of (2), we propose the following assumptions:

ξ is an ℱT measurable square integrable random variable;

the obstacle {St:0≤t≤T} is an ℱt-progressive measurable continuous real valued process which satisfies Esup0≤t≤T(St)2<∞. We always assume that ST≤ξ, a.s.;

f(·,y,z) and gj(·,y,z) are two progressive measurable functions such that, for any t∈[0,T], y1,y2∈ℝ, z1,z2∈ℝd,

f(s,·,·) is continuous and |f(s,y,z)|≤M(1+|y|+|z|);

E∫0T|f(t,0,0)|2dt<∞, ∑j=1∞E∫0T|gj(t,0,0)|2dt<∞;

|f(s,y1,z1)-f(s,y2,z2)|2≤C(|y1-y2|2+|z1-z2|2), |gj(s,y1,z1)-gj(s,y2,z2)|2≤Cj|y1-y2|2+αj|z1-z2|2, where M, C, Cj, and αj are nonnegative constants with ∑j=1∞Cj<∞ and α=∑j=1∞αj<1.

3. Main Result

In order to get the solution of (2), we consider the following RBSDEs driven by finite Brownian motions:
(3)Yt=ξ+∫tTf(s,Ys,Zs)ds+∑j=1n∫tTgj(s,Ys,Zs)dβj(s)-∫tTZsdW(s)+KT-Kt,0≤t≤T.

Firstly, we consider a special case of (3); that is, the functions f and g do not depend on (Y,Z):
(4)Yt=ξ+∫tTf(s)ds+∑j=1n∫tTgj(s)dβj(s)-∫tTZsdW(s)+KT-Kt,0≤t≤T,n≥1.

We will get the existence and uniqueness of the solution of (4) by means of Snell envelope and martingale representation theorem.

Theorem 2.

Assume that (H1)-(H2), f∈ℋ2, g∈ℋ2. Then, there exists a triple (Yt,Zt,Kt)0≤t≤T∈𝒮2×ℋ2×𝒜2 which is a solution of (4).

Proof.

Let
(5)𝒞t=ℱtW⋁(⋁j=1nℱt,Tβj),
and we define η={ηt}0≤t≤T as
(6)ηt=ξ1{t=T}+St1{t<T}+∫0tf(s)ds+∑j=1n∫0tgj(s)dβj(s).
Then, η is 𝒞t-adapted continuous process; furthermore;
(7)sup0≤t≤T|ηt|∈L2(Ω).
So, the Snell envelope of η is given by
(8)St(η)=esssupν∈𝒯E[ην∣𝒞t],
where 𝒯 is the set of all 𝒞t stopping time such that 0≤ν≤T.

By the definition of η, we can deduce that
(9)E[sup0≤t≤T|St(η)|2]<∞.

Due to the Doob-Meyer decomposition, we have
(10)St(η)=E[∑j=1nξ+∫0Tf(s)ds+∑j=1n∫0Tgj(s)dβj(s)+KT∣𝒞t]-Kt,
where {Kt}0≤t≤T is a 𝒞t-adapted, continuous, and nondecreasing process such that K0=0 and EKT2<∞. So, we have
(11)E[+∑j=1n∫0Tgj(s)dβj(s)+KT∣𝒞t]|2sup0≤t≤T|E[∑j=1nξ+∫0Tf(s)ds+∑j=1n∫0Tgj(s)dβj(s)+KT∣𝒞t]|2]<∞.

Martingale representation theorem yields that there exists 𝒞t-progressive measurable process {Zt}∈ℝd such that
(12)Mt≜E[ξ+∫0Tf(s)ds+∑j=1n∫0Tgj(s)dβj(s)+KT∣𝒞t]=M0+∫0tZsdW(s),0≤t≤T.

Let Yt=esssupν∈𝒯E[ξ1{ν=T}+Sν1{ν<T}+∫tνf(s)ds+∑j=1n∫tνgj(s)dβj(s)∣𝒞t]; then,
(13)Yt+∫0tf(s)ds+∑j=1n∫0tgj(s)dβj(s)=St(η)=Mt-Kt=M0+∫0tZsdW(s)-Kt,0≤t≤T.
Therefore,
(14)Yt=ξ+∫tTf(s)ds+∑j=1n∫tTgj(s)dβj(s)-∫tTZsdW(s)+KT-Kt.

By the definitions of Yt and St(η), ξ≥ST,
(15)Yt+∫0tf(s)ds+∑j=1n∫0tgj(s)dβj(s)=St(η)≥ηt=ξ1{t=T}+St1{t<T}+∫0tf(s)ds+∑j=1n∫0tgj(s)dβj(s)≥ST1{t=T}+St1{t<T}+∫0tf(s)ds+∑j=1n∫0tgj(s)dβj(s).
So, we have Yt≥St.

Finally, from Hamadène [15], we get ∫0T(St(η)-ηt)dKt=0; that is,
(16)∫0T(Yt-St)dKt=0.
It shows that the process (Yt,Zt,Kt)0≤t≤T is a solution of (4).

Theorem 3.

Under the assumptions of (H1)–(H3), there exists a unique solution (Yt,Zt,Kt)0≤t≤T of (3).

Proof.

Let 𝒫=𝒮2×ℋ2 be endowed with the norm
(17)∥(Y,Z)∥β=(E[∫0Teβs(|Ys|2+|Zs|2)ds])1/2
for a suitable constant β>0. We define the map Φ from 𝒫 into itself and (Y~,Z~) and (Y′~,Z′~) are two elements of 𝒫. Define (Y,Z)=Φ(Y~,Z~), (Y′,Z′)=Φ(Y′~,Z′~), where (Y,Z,K) and (Y′,Z′,K′) are solutions of (4) associated with (ξ,f(t,Y~,Z~),gj(t,Y~,Z~),S), and (ξ,f(t,Y′~,Z′~),gj(t,Y′~,Z′~),S′), respectively. Set (Y¯,Z¯)=(Yt-Yt′,Zt-Zt′) and
(18)ΨM(x)=x21{-M≤x≤M}+M(2x-M)1{x>M}-M(2x+M)1{x<-M}.
If we define ΨM′(x)/x=2, when x=0, then, 0≤ΨM′(Ys¯)/Ys¯≤2. Applying Itô formula to eβtΨM(Ys¯), we have
(19)eβtΨM(Yt¯)+β∫tTeβsΨM(Ys¯)ds+∫tTeβs1{-M≤Ys¯≤M}|Z¯s|2ds=∫tTeβsΨM′(Ys¯)(f(s,Ys~,Zs~)-f(s,Ys′~,Zs′~))ds+∑j=1n∫tTeβs1{-M≤Ys¯≤M}|gj(s,Ys~,Zs~)-gj(s,Ys′~,Zs′~)|2ds-∑j=1n∫tTeβsΨM′(Ys¯)(gj(s,Ys~,Zs~)-gj(s,Ys′~,Zs′~))dβj(s)-∫tTeβsΨM′(Ys¯)Z¯sdW(s)+∫tTeβsΨM′(Ys¯)(dKs-dKs′).
Taking expectation on both sides of (19) and noticing that ∫tTeβsΨM′(Ys¯)(dKs-dKs′)≤0, we have
(20)EeβtΨM(Yt¯)+Eβ∫tTeβsΨM(Ys¯)ds+E∫tTeβs1{-M≤Ys¯≤M}|Z¯s|2ds≤E∫tTeβsΨM′(Ys¯)(f(s,Ys~,Zs~)-f(s,Ys′~,Zs′~))ds+∑j=1nE∫tTeβs1{-M≤Ys¯≤M}|gj(s,Ys~,Zs~)-gj(s,Ys′~,Zs′~)|2ds≤2E∫tTeβsYs¯(f(s,Ys~,Zs~)-f(s,Ys′~,Zs′~))ds+∑j=1nE∫tTeβs|gj(s,Ys~,Zs~)-gj(s,Ys′~,Zs′~)|2ds≤2C1-αE∫tTeβs|Ys¯|2ds+(∑j=1∞Cj+1-α2)E∫tTeβs|Ys~-Ys′~|2ds+1+α2E∫tTeβs|Zs~-Zs′~|2ds.
Let γ=2C/(1-α), C¯=2(∑j=1∞Cj+((1-α)/2))/(1+α), β=γ+C¯, and M→∞; we have
(21)C¯E∫tTeβs|Ys-Ys′|2ds+E∫tTeβs|Zs-Zs′|2ds≤1+α2E∫tTeβs(C¯|Ys~-Ys′~|2+|Zs~-Zs′~|2);
that is,
(22)∥(Ys,Zs)∥β2≤1+α2∥(Ys′,Zs′)∥β2.

It follows that Φ is a strict contraction on 𝒫 with the norm ∥·∥β, where β is defined as above. Then, Φ has a fixed point (Y,Z,K) which is the unique solution of (4) from the Burkholder-Davis-Gundy inequality.

With all the preparations, we will give the main result of this paper as follows.

Theorem 4.

Under the conditions of (H1)–(H3), there exists a unique solution (Yt,Zt,Kt)0≤t≤T∈𝒮2×ℋ2×𝒜2 of (2).

Proof (existence).

By Theorem 3, for any n≥1, there exists a unique solution of (3), denoted by (Ytn,Ztn,Ktn),
(23)Ytn=ξ+∫tTf(s,Ysn,Zsn)ds+∑j=1n∫tTgj(s,Ysn,Zsn)dβj(s)-∫tTZsndW(s)+KTn-Ktn.

In the following parts, we will claim that (Ytn,Ztn,Ktn) is a Cauchy sequence in 𝒮2×ℋ2×𝒜2. Without loss of generality, we let n<m. Applying general Itô formula to |Ytn-Ytm|2, we have. (24)|Ytn-Ytm|2+∫tT|Zsn-Zsm|2ds=2∫tT(Ysn-Ysm)(f(s,Ysn,Zsn)-f(s,Ysm,Zsm))ds+∑j=n+1m∫tT|gj(s,Ysn,Zsn)-gj(s,Ysm,Zsm)|2ds-2∑j=n+1m∫tT(Ysn-Ysm)(gj(s,Ysn,Zsn)-gj(s,Ysm,Zsm))dβj(s)-2∫tT(Ysn-Ysm)(Zsn-Zsm)dW(s)+2∫tT(Ysn-Ysm)(dKsn-dKsm).
Taking expectation on both sides of (24) and noting that ∫tT(Ysn-Ysm)(dKsn-dKsm)≤0, we obtain
(25)E|Ytn-Ytm|2+E∫tT|Zsn-Zsm|2ds≤2E∫tT(Ysn-Ysm)(f(s,Ysn,Zsn)-f(s,Ysm,Zsm))ds+∑j=n+1mE∫tT|gj(s,Ysn,Zsn)-gj(s,Ysm,Zsm)|2ds.
By (H3) and elementary inequality 2ab≤βa2+(1/β)b2, β>0, we obtain
(26)E|Ytn-Ytm|2+E∫tT|Zsn-Zsm|2ds≤2C1-αE∫tT|Ysn-Ysm|2ds+1-α2E∫tT|Ysn-Ysm|2ds+1-α2E∫tT|Zsn-Zsm|2ds+αE∫tT|Zsn-Zsm|2ds+[∑j=n+1mCj]E∫tT|Ysn-Ysm|2ds.
Furthermore,
(27)E|Ytn-Ytm|2+1-α2E∫tT|Zsn-Zsm|2ds≤CpE∫tT|Ysn-Ysm|2ds,
where Cp=(2C/(1-α))+((1-α)/2)+∑j=n+1mCj.

By Gronwall’s inequality and Burkholder-Davis-Gundy inequality, we have
(28)E[sup0≤t≤T∫tT|Ysn-Ysm|2ds]⟶0.

Denote the limit of (Ytn,Ztn,Ktn) by (Yt,Zt,Kt); we will show that (Yt,Zt,Kt) satisfies (2). If it is necessary, we can choose a subsequence of (3). By Hölder’s inequality,
(29)E|∫tT(f(s,Ys,Zs)-f(s,Ysn,Zsn))ds|2≤TE∫tT|(f(s,Ys,Zs)-f(s,Ysn,Zsn))|2ds⟶0.
From (27), we know
(30)E∫0T|Ytn-Yt|2dt⟶0,
and Ytn→Yt, a.e., so
(31)E∫0T|Ytn+1-Ytn|2dt≤12n.
For any n,
(32)|Ytn|≤|Yt1|+∑i=1n-1|Yti+1-Yti|≤|Yt1|+∑i=1∞|Yti+1-Yti|.
Then, we have
(33)E∫0Tsupn|Ytn|2dt≤E∫0T(|Yt1|+∑i=1∞|Yti+1-Yti|)2dt≤E∫0T|Yt1|2dt+∑i=1∞E∫0T|Yti+1-Yti|2dt≤E∫0T|Yt1|2dt+∑i=1∞12i.
From (H4), it follows
(34)E∫0Tsupn|f(s,Ys,Zs)-f(s,Ysn,Zsn)|2ds≤2CE∫0T(supn|Ysn|2+|Ys|2+supn|Zsn|2+|Zs|2)ds<∞.
Applying Lebesgue dominated convergence theorem, we deduce that (Yt,Zt,Kt) is the solution of (2) by continuity of the functions f and g.

Uniqueness. Let (Yti,Zti,Kti) (i=1,2) be two solutions of (2), Y¯t=Yt1-Yt2, Z¯t=Zt1-Zt2. We apply Itô formula to eβtΨM(Y¯t), for any β∈ℝ,
(35)eβtΨM(Y¯t)+β∫tTeβsΨM(Y¯s)ds+∫tTeβs1{-M≤Y¯s≤M}|Z¯s|2ds=∫tTeβsΨM′(Y¯s)(f(s,Ys1,Zs1)-f(s,Ys2,Zs2))ds+∑j=1∞∫tTeβs1{-M≤Y¯s≤M}|gj(s,Ys1,Zs1)-gj(s,Ys2,Zs2)|2ds-∑j=1∞∫tTeβsΨM′(Ys¯)×(gj(s,Ys1,Zs1)-gj(s,Ys2,Zs2))dβj(s)-∫tTeβsΨM′(Ys¯)Z¯sdWs+∫tTeβsΨM′(Ys¯)(dKs1-dKs2).
Taking expectation on both sides of (35),
(36)EeβtΨM(Y¯t)+βE∫tTeβsΨM(Y¯s)ds+E∫tTeβs1{-M≤Y¯s≤M}|Z¯s|2ds≤2E∫tTeβsY¯s(f(s,Ys1,Zs1)-f(s,Ys2,Zs2))ds+∑j=1∞E∫tTeβs1{-M≤Y¯s≤M}|gj(s,Ys1,Zs1)-gj(s,Ys2,Zs2)|2ds≤(2C1-∑j=1∞αj+∑j=1∞Cj+1-∑j=1∞αj2)E∫tTeβs|Ys¯|2ds+1+∑j=1∞αj2E∫tTeβs|Z¯t|2ds.
Let M→∞, and applying monotone convergence theorem, we have
(37)Eeβt|Y¯t|2+(β-2C1-α-∑j=1∞Cj-1-α2)×E∫tTeβs|Y¯s|2ds+1-α2E∫tTeβs|Z¯s|2ds≤0.

When β is taken sufficiently large, we have Y¯t=0, a.e., for all s∈[t,T]. So, we have Z¯t=0, a.e. Then, we complete the proof.

Acknowledgments

The authors would like to take this chance to express their sincere gratitude to the National Natural Science Foundation of China (11201004 and 11371029), Natural Science Foundation of Anhui Province (KJ2011B176 and KJ2013B288), Professors(Doctors) Scientific Research Foundation of Suzhou University (2013jb04), and Foundation of Laboratory of Intelligent Information Processing of Suzhou University (2010YKF11).

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