Two problems for stochastic flows associated with nonlinear parabolic equations

In this paper, we solve two problems for some nonlinear SPDE driven by Fisk-Stratonovich stoachastic integral. The main assumption is the commuting property of the drift and diffusion vector fields with respect of the Lie bracket. In the first problem (P1) we construct a classical solution for some nonlinear SPDE of parabolic type by assuming the compatibilty condition concerning the mentioned vector fields. The second problem (P2) is a related filtering one for a non-markovian system of SDEs involving a backward parabolic equation of Kolmogorov type with parameter.