  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.
