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Two problems for stochastic flows associated with nonlinear parabolic equations. (English) Zbl 1389.60082
Summary: 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.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
35F21 Hamilton-Jacobi equations
35R60 PDEs with randomness, stochastic partial differential equations