Diffusion Processes and Partial Differential Equations

拟开展的活动有三个主要领域。第一个是发展一种基于伊藤方程解的拟导数的技术，并将其应用于考虑到第一次从区域中退出的过程。第二个领域是研究在过滤问题和测值过程理论中出现的SPDEs解的光滑性，以便能够保证逼近到其解的一定的收敛速度。第三个领域是估计退化控制扩散过程数值逼近的收敛速度。该项目涉及对某些对象的概率行为的调查。第一部分的目的是为了更好地理解与随机过程的最优控制有关的平均量，这些随机过程出现在从金融到航空航天工程的所有类型的应用中。第二部分针对与消除友军火力的图像重建或高性能计算等许多实际问题直接相关的问题。它在处理细菌种群进化等问题时也是至关重要的，这些问题在生物技术中可能是重要的。第三部分讨论了在上述应用中出现的随机过程最优控制的实际问题的解决方法。

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.