The Convergence Property Of The Reflected Backward Stochastic Differential Equations

In this paper, we mainly discuss the reflected backward stochastic differential equations, and give the convergence results of the solutions when the parameters of the equation converge under some conditions.In 1990, E.Pardoux and Peng gave the nonlinear backward stochastic differential equation (BSDE in short) which is the fundamental article [1]. Afterward Peng (see[2]) introduced the concept of a supersolution of BSDE with generator, which needs to set a RCLL increasing process. In this paper he gave us the monotonic limit theorem of BSDE, i.e: if a sequence of RCLL supersolutions of a backward stochastic differential equations converges monotonically, then the limit is also a supersolution under some conditions. EL-Karoui et al (see [4]) studied one kind of the reflected BSDE with one barrier. This kind of equation previously set an " obstacle " S_t. They proved the existence and uniqueness of the triple solution (y_t,,z_t,K_t). One part of the solutions is a increasing process K_t which pushes the solution y_t upwards and also requires the push power to be minimum. This kind of the reflected solutions of BSDE's is different from the solutions of normal BSDE's and the supersolution in [2]. In this paper, we consider the convergence property of the reflected solutions of BSDE's.First we consider a sequence of the reflected BSDEs which parameters are (fⁱ,ξⁱ, S) for finite horizon (i = 1,2,3, …) :K_tⁱ is continuous and increasing,K₀ⁱ = 0,and ∫₀^T(y_tⁱ- S_t)dK_tⁱ = 0.We obtain the convergence property of the reflected solutions of BSDE's when the above equation satisfies some conditions, that is the main part of our paper:Theorem 3.1 When i→∞, we assume (fⁱ, ξⁱ, S) → (f, ξ, S), where ξⁱ converges inï¿¡2, and is uniformly bounded, for each t 6 [0, T], y € R, z € Rd, /*(ï¿¡,?/, z) converges to f(t,y,z) in Af2 , here /'(i.OjO) is uniformly bounded about i in A42. Then for the solution of the equation, we have: y\ —> yt (in ?S2), at the same time there exist zt G M.2 and a increasing process Kt, 0 < t < T, such that : a triple solution (yt,zt,Kt) is the solution of the following equation which paramaters are (/, ï¿¡, S):Vt = Z + Sl /(*> V.. *.)<*? + KT-Kt- tf(zs, dBs), 0Su 0<_t oo, ï¿¡' converges to ï¿¡ in ï¿¡2(f2, T, P), and for each t € [0, oo], y e R, z € Rd, /0°° /'(s, ys, i"s)rfs converges to /0°° f(s, ys, zs)ds in C2{Q,, T, P). At the same time , ï¿¡*, /0°° /l(s, 0,0)ds is uniformly bounded about i in C2, and V t € [0, oo], y\ is uniformly bounded about i in yt (in <52), at the same time there exist zt € M2 and a increasing process Kt, 0 < t < oo, such that : a triple solution (yt, zt, Kt) is the solution of the following equation which paramaters are (/, ï¿¡, 5):Vt = e + /t°° /(a, &, ?.)rf5 + i^oo - Kt - ft°°(z3, dBa), t e [0, oo]; Vt>St, *6[0,oo]; (7)Kt is continuous and increasing, KQ = 0, and/0°c(yt — St)dKt = 0.Here zt is the strong limit of z\ in X2, and for each t, Kt is the limit of K\ in ï¿¡2.At the last part of our paper, we will further extend our problem. We consider the convergence property of reflected BSDE's with upper and lower barriers (see [6]), and obtain our Theorem 5.2Theorem 5.2 When i -?? oo, we assume (f,C,S,U) -> (ftï¿¡,S,U), where f converges in ï¿¡2, and is uniformly bounded, for each t yt (in 52), atthe same time there exist zt € M2 and increasing processes Kt, At, 0 < t < T, such that : a quadruple solution (yt, zt,Kt,At) is the solution of the following equation which paramaters are (/, ï¿¡, S, U):Vt = Z + Jf f{s, ys, zs)ds + KT-Kt- {AT - At) - tf(z3, dB3), 0 < t < T; St