Backward Doubly SDEs and Semilinear Stochastic PDEs in a convex domain

TL;DR

This paper establishes existence and uniqueness for reflected backward doubly stochastic differential equations in convex domains and provides a probabilistic interpretation of related semilinear stochastic PDEs using a stochastic flow approach.

Contribution

It introduces new existence and uniqueness results for RBDSDEs in convex domains and links them to semilinear SPDEs through a stochastic flow method.

Findings

Existence and uniqueness of solutions for RBDSDEs in convex domains.

Probabilistic representation of semilinear SPDEs via RBDSDEs.

Solution characterized by a process in Sobolev space and a measure controlled by boundary behavior.

Abstract

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDDSEs) in a convex domain D. 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 m 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.