Some related problems for stochastic partial differential equations and backward stochastic differential equations

PI将研究三种类型的随机偏微分方程（SPDEs）和倒向随机微分方程（BSDEs）问题。首先，他将建立spde和bsde之间的连接；并将用它来证明这类spde解的强唯一性和正则性。他还将利用大偏差原理研究金融市场的罕见事件现象。其次，PI将导出一个随机极大原理（SMP）的随机最优控制问题产生于实际应用中，其中系统的系数是不光滑的。这将导致两个新的bsde类。他将研究它们的解，并探索它们的数值近似。最后，PI将考虑基于部分信息的参数估计问题。证明这种估计的一致性的关键在很大程度上依赖于滤波稳定性结果的建立。本研究的动力来自于对某些人口模型、无线通信和数学金融的研究。PI将开发数学工具来研究由这些问题产生的spde和bsde。所得结果将应用于各种现实世界的应用，以造福社会。例如，SMP可以应用于电力控制问题，以提高通信网络的效率，为社会服务。对金融市场的研究有助于政府对市场的监管；特别是，大偏差问题将有助于金融市场事件的特征化。参数估计问题可用于包括污染源在内的各种目标的定位。对研究生和研究生在随机分析和最优控制方面的培养也有长期的影响。

The PI will study three types of problems about stochastic partial differential equations (SPDEs) and backward stochastic differential equations (BSDEs). Firstly, he will build a connection between SPDEs and BSDEs; and will use it to prove the strong uniqueness and regularity properties for the solutions to such SPDEs. He will also study the rare eventsphenomena for financial market using large deviation principle.Secondly, the PI will derive a stochastic maximum principle (SMP) for stochastic optimal control problems arising from real world applications where the coefficients of the systems are not smooth. This will lead to two new classes of BSDEs. He will study their solutions and will explore the numerical approximation for these. Finally,the PI will consider the parameter estimationproblem based on partial information. The key to the proof of theconsistency of such estimates is heavily dependent on theestablishment of a filtering stability result.This research is motivated from the study of certain populationmodels, the wireless communication and the mathematical finance. The PI will develop mathematic tools to study SPDEs and BSDEs arising from these problems. The results obtained will be applied to various real world applications for the benefitto the society. For example, the SMP can be applied to the power controlproblem to serve the society in improving the efficiency of thecommunication network. The study of the financial market will helpthe government to regulate the market; especially, the largedeviation problem will help to characterizerare events in the financial market. The parameter estimationproblem can be used to locate various targets including the hiddensource of pollution. Long-term effects on graduate and post-graduate training of students in stochastic analysis and optimal control are also expected.