Backward Stochastic Partial Differential Equations And Their Applications

The dissertation is concerned with the existence, uniqueness and regularity of solutions to semi-linear backward stochastic PDEs. It addresses the Cauchy problem for the equations and the Dirichlet boundary problem under the super-parabolicity condition.The first two chapters are preliminaries. In Chapter 3, the existence and uniqueness of Hm-solutions for super-parabolic backward stochastic PDEs in the whole space are proved.In Chapter 4, the existence and uniqueness of strong solutions to the Dirichlet problem for super-parabolic backward stochastic PDEs are proved in C2 domains, and then the regularity of solutions to linear equations are studied. The proofs make use of the decomposition and flat-tening techniques as well as the method of parameter continuity. As by practices, a generalized Ito formula and a comparison theorem for weak solutions are obtained.In Chapter 5, degenerate parabolic backward stochastic PDEs are studied in the whole space. The existence and uniqueness of weak solutions to linear equations are proved, and the Wm,p estimates (p?2) for the solution w.r.t. the spatial variables are established. Moreover, the existence and uniqueness results are proved for the semi-linear equations with the Lipschitz conditions, and the regularity of the solution is also discussed.In Chapter 6, a backward stochastic PDE in the bounded domain with quadratic growth is studied for the first time. The existence and uniqueness are proved separately under different conditions.Finally, in Chapter 7, three simple examples are given to illustrate the above results. They are the first exit times of non-Markovian processes, the stochastic Feynman-Kac formula and the the Pontryagin maximum principle for degenerate partially observed optimal control prob-lem.