Diffusion Processes and Partial Differential Equations

9876586拟议的活动主要有三个领域。第一个是发展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1部分旨在更好地理解与随机过程的最优控制相关的平均量。第2部分的目标是与许多实际问题直接相关的问题，例如图像重建或消除“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。第三部分讨论了解决随机过程最优控制问题的实际方法，例如在金融中出现的问题。

9876586There are three main areas of proposed activity. The first one is developing the technique based on quasiderivatives of solutions of Ito's equations and applying it for 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 investigations of probabilistic behavior of certain objects. Part 1 is aimed at a better understanding of averaged quantities related to optimal control of random processes. Part 2 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 3 deals with practical ways of solving problems of optimal control of random processes which arise for instance in finance.