Mathematical Sciences: Diffusion Processes and Partial Differential Equations

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9625483 Krylov ABSTRACT The project concentrates on three areas: probabilistic solutions of PDEs, stochastic partial differential equations, and numerical solutions of control problems for diffusion processes. In the first area, the objective is to further develop the technique based on quasiderivatives mf solutions of Ito's equations and apply it for processes considered up to the first exit time from a domain. In the second area, it is to investigate smoothness of solutions of SPDEs arising in filtering problems and population genetics in order to be able to guarantee certain rates of convergence of approximations to their solutions. In the third area, the aim is to get estimates of the rate of convergence. The project relates to investigations of probabilistic behavior of certain objects. Part 1 is aimed at better understanding of averaged quantities and is thought of as been important for parts 2 and 3. Part 2 aims at problems directly related to many practical issues such as image reconstruction or high-performance. It is also vital in dealing with problems like evolution of bacteria populations, which may be important in biotechnology. Part 3 deals with problems of optimal control of random processes which arise, for instance, in finance.