An implicit numerical scheme for a class of backward doubly stochastic differential equations

Abstract

In this paper, we consider a class of backward doubly stochastic differential equations (BDSDE for short) with general terminal value and general random generator. Those BDSDEs do not involve any forward diffusion processes. By using the techniques of Malliavin calculus, we are able to establish the L p -Hölder continuity of the solution pair. Then, an implicit numerical scheme for the BDSDE is proposed and the rate of convergence is obtained in the L p -sense. As a by-product, we obtain an explicit representation of the process Y in the solution pair to a linear BDSDE with random coefficients.2010 Mathematics Subject Classification. 60H10; 60H07; 60H05. Key words and phrases. Malliavin calculus, Backward doubly stochastic differential equations, explicit solution to linear bdsde, implicit scheme, Hölder continuity of the solution pairs, rate of convergence. D.

Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps

“…FBDSDEs have also shown relevance in optimal filtering problems, as observed in [6]. Finally, it is worth mentioning that numerical investigations in [7] and [8], employing Malliavin calculus for FBSDEs and BDSDEs, respectively, have led to promising research on a linear version of FBDSDEJ (1.1) in both finite and infinite dimensions.…”