Mathematical Sciences: Diffusion Processes and Partial Differential Equations

摘要本项目主要研究三个领域：偏微分方程的概率解、随机偏微分方程和扩散过程控制问题的数值解。在第一个领域，目标是进一步发展基于Ito方程的拟导数mf解的技术，并将其应用于考虑到从一个域的第一次退出时间的过程。在第二个领域，它是研究在滤波问题和群体遗传学中产生的spde解的平滑性，以便能够保证其解的近似收敛率。在第三个领域，目标是得到收敛速度的估计。这个项目涉及对某些物体的概率行为的调查。第1部分旨在更好地理解平均量，并且被认为对第2部分和第3部分很重要。第2部分针对的问题直接关系到许多实际问题，如图像重建或高性能。它在处理细菌种群进化等问题时也至关重要，这在生物技术中可能很重要。第3部分讨论随机过程的最优控制问题，例如在金融领域。

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.