In this paper we establish a probabilistic representation for the spatial gradient ofthe viscosity solution to a quasilinear parabolic stochastic partial differential equations(SPDE, for short) in the spirit of the Feynman-Kac formula, without using thederivatives of the coefficients of the corresponding backward doubly stochastic differentialequations (FBDSDE, for short).
"Representation theorems for SPDEs via backward doubly." Electron. Commun. Probab. 18 1 - 15, 2013. https://doi.org/10.1214/ECP.v18-2223