Diffusion Processes and Partial Differential Equations

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There are three main areas of proposed activity. The first one is developing a technique based on quasiderivatives of solutions of Ito's equations and applying it to processes considered up to the first exit time from domains. The second area is investigating smoothness of solutions of SPDEs arising in filtering problems and the theory of measure--valued processes in order to be able to guarantee certain rates of convergence of approximations to their solutions. The third area is estimating the rate of convergence of numerical approximations for degenerate controlled diffusion processes. The project relates to the investigation of the probabilistic behavior of certain objects. Part one is aimed at better understanding of averaged quantities related to optimal control of random processes arising in all types of applications from finance to aerospace engineering. Part two aims at problems directly related to many practical issues such as image reconstruction or high-performance computing in eliminating "friendly fire". It is also vital in dealing with problems like evolution of bacteria population which may be important in biotechnology. Part three deals with ways of solving practical problems of optimal control of random processes which arise in the applications mentioned above.