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Backward stochastic partial differential equations driven by infinite-dimensional martingales and applications. (English) Zbl 1186.60049
Summary: This paper studies first a result of existence and uniqueness of the solution to a backward stochastic differential equation driven by an infinite-dimensional martingale. Then, we apply this result to find a unique solution to a backward stochastic partial differential equation in infinite dimensions. The filtration considered is an arbitrary right-continuous filtration, not necessarily the natural filtration of a Wiener process. This, in particular, allows us to study more applications, for example, the maximum principle for a controlled stochastic evolution system. Some examples are discussed in the paper as well.
MSC:
60H10Stochastic ordinary differential equations
60H15Stochastic partial differential equations
60G44Martingales with continuous parameter
34F05ODE with randomness
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)