Backward Doubly SDEs and Semilinear Stochastic PDEs in a convex domain

Matoussi Anis, Sabbagh Wissal

Published 2014-05-22, updated 2014-10-13Version 2

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDDSEs) in a convex domain D without any regularity conditions on the boundary. Moreover, using a stochastic flow approach a probabilistic interpretation for a class of reflected SPDE's in a domain is given via such RBDSDEs. The solution is expressed as a pair (u,{\nu}) where u is a predictable continuous process which takes values in a Sobolev space and {\nu} is a random regular measure. The bounded variation process K, component of the solution of the reflected BDSDE, controls the set when u reaches the boundary of D. This bounded variation process determines the measure m from a particular relation by using the inverse of the flow associated to the the diffusion operator.