Backward doubly SDEs and semilinear stochastic PDEs in a convex domain

Anis Matoussi (1) , Wissal Sabbagh (1) , Tusheng Zhang (2)

1 LMM - Laboratoire Manceau de Mathématiques
2 School of Mathematics - Department of Mathematics [Manchester]

This paper presents existence and uniqueness results for reflected backward doubly stochastic differential equations (in short RBDSDEs) in a convex domain D without any regularity conditions on the boundary. Moreover, using a stochastic flow approach a probabilistic interpretation for a system of reflected SPDEs in a domain is given via such RBDSDEs. The solution is expressed as a pair (u, ν) where u is a predictable continuous process which takes values in a Sobolev space and ν is a random regular measure. The bounded variation process K, the 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 ν from a particular relation by using the inverse of the flow associated to the the diffusion operator.

Keywords: and phrases: Stochastic Partial Differential Equation; Reflected Back-; ward Doubly Stochastic Differential Equation; Skorohod Problem; Convex Domains; Stochastic Flow; Flow of Diffeomorphisms; Regular Measure

Domains: Probability [math.PR]; Optimization and Control [math.OC]; Analysis of PDEs [math.AP]

Anis Matoussi, Wissal Sabbagh, Tusheng Zhang. Backward doubly SDEs and semilinear stochastic PDEs in a convex domain. Stochastic Processes and their Applications, 2017, 127 (9), pp.2781 - 2815. ⟨10.1016/j.spa.2016.12.010⟩. ⟨hal-01740652⟩