Representation theorems for backward doubly stochastic differential equations

Auguste Aman (1)

1 LMAI - Laboratoire de Mathématiques Appliquées et Informatique

In this paper we study the class of backward doubly stochastic differential equations (BDSDEs, for short) whose terminal value depends on the history of forward diffusion. We first establish a probabilistic representation for the spatial gradient of the stochastic viscosity solution to a quasilinear parabolic SPDE in the spirit of the Feynman-Kac formula, without using the derivatives of the coefficients of the corresponding BDSDE. Then such a representation leads to a closed-form representation of the martingale integrand of BDSDE, under only standard Lipschitz condition on the coefficients.

Mots clés: Adapted solution; anticipating stochastic calculus; backward doubly SDEs; stochastic partial dfferential equation; stochastic viscosity solutions.; stochastic viscosity solutions

Domaines: Probabilités [math.PR]

Auguste Aman. Representation theorems for backward doubly stochastic differential equations. 2008. ⟨hal-00196841v4⟩