Diffusion Processes and Partial Differential Equations

拟议的活动主要有三个领域。第一个是开发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从域的第一次退出时间的过程。第二个领域是研究滤波问题中产生的spde解的平滑性和测量值过程理论，以便能够保证其解的近似收敛率。第三个领域是估计退化控制扩散过程的数值近似的收敛速度。这个项目涉及对某些物体的概率行为的调查。第一部分旨在更好地理解与从金融到航空航天工程的所有类型应用中出现的随机过程的最佳控制相关的平均量。第二部分的目标是直接涉及许多实际问题的问题，如图像重建或消除“友军火力”的高性能计算。它在处理细菌种群进化等问题时也是至关重要的，这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.