| Since Non-linear backward stochastic differential equations (BSDEs in short) were introduced by Pardoux and Peng [1], the theory designed by this paper has been found to be powerful tool for obtaining the probabilistic in-terpretation of partial differential equation (PDE for short) while backward doubly stochastic differential equations (BDSDEs for short) can obtain the probabilistic interpretation of stochastic PDEs.On the other hand in order to give a probabilistic interpretation to the particularly class of stochastic PDEs, the solution of which has some limits, Yufeng Shi and Yanling Gu [29] introduced a new kind of equation:reflected backward doubly stochastic differential equations (RBDSDEs for short). Au-guste Aman and Modeste N'Zi [5] have done the similar research.However, RBDSDEs can not be solved explicitly. To develop numerical method and numerical algorithm is very helpful,theoretically and practically.In this paper we study two numerical forms (implicit explicit) of approx-imating solutions of reflected backward doubly stochastic differential equa-tions (RBDSDEs for short) with one lower continuous barrier and prove its convergence.Consider the following RBDSDEs with one lower barrierwhere Bt and Wt are two mutually independent standard Brown motion processes. Kt is an increasing process and K0 = 0. We divide the time in-terval [O,T] into n parts:O= t0<t1<...< tn=T,δ:= tj-tj-1= T/n,for1≤j≤n. Now we define the scaled random walk{B.n},{W.n},by setting B0n=W0n=0, where are two mutually independent Bernoulli sequences,whichOn the small interval [tj,tj+1], the equation above can be writen asBy penalization method, replace we haveTaking conditional expectation, We solve the equation above and obtain the implicit solution. In fact, ytp,n can not be solved explicitly in many cases which f nonlinearly depends on y. In this case, we apply the explicit penaliza-tion scheme replacing yJp,n in f by E[(?)].Based on the weak convergence of filtrations, we prove the convergence of both implicit and explicit scheme.The paper is organized as follows:The first chapter introduces the current research of RBDSDEs and its nu-merical computation.In chapter two, we offer some preliminaries of RBDSDEs.In chapter three, by replacing Bernoulli sequences and penalization method, we consturct the implicit and explicit scheme of RBDSDEs.In chapter fore, based on the weak convergence of filtrations, we prove the convergence of both implicit and explicit scheme. |
