Representation theorems for backward doubly stochastic differential equations

Details Representation theorems for backward doubly stochastic differential equations Representation theorems for backward doubly stochastic differential equations

2007-12-13

arxiv.org/abs/0712.2219v4

Mathematics

Probability

The version of this article have 20 pages and is submitted to Journal Bernoulli for publication

Scientific paper

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.

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