Mathematical Sciences: Diffusion Processes and Partial Differential Equations

9625483克雷洛夫摘要 该项目集中在三个领域：概率解的偏微分方程，随机偏微分方程，扩散过程的控制问题的数值解。在第一个领域，我们的目标是进一步发展的技术基础上quasidifferentiators MF解决方案的伊藤方程，并将其应用于过程中考虑到第一次退出时间从一个域。 在第二个领域，它是调查平滑的解决方案所产生的SPDE过滤问题和人口遗传学，以便能够保证一定的速度收敛的近似解决方案。在第三个方面，目标是得到收敛速度的估计。 该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，被认为对第2部分和第3部分很重要。第2部分针对与许多实际问题直接相关的问题，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部分涉及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.