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 events phenomena 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 estimation problem based on partial information. The key to the proof of the consistency of such estimates is heavily dependent on the establishment of a filtering stability result.
This research is motivated from the study of certain population models, 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 benefit to the society. For example, the SMP can be applied to the power control problem to serve the society in improving the efficiency of the communication network. The study of the financial market will help the government to regulate the market; especially, the large deviation problem will help to characterize rare events in the financial market. The parameter estimation problem can be used to locate various targets including the hidden source of pollution. Long-term effects on graduate and post-graduate training of students in stochastic analysis and optimal control are also expected.
