Special Session 24: SPDEs/SDEs and Stochastic Systems with Control/Optimization and Applications

A Unified System of SPDEs with Levy Jumps vs. Stochastic Differential Games

Wanyang Dai
Nanjing University
Peoples Rep of China
We study a unified system of stochastic partial differential equations (SPDEs) with Levy jumps in a forward-backward coupling manner. The partial differential operators in its drift, diffusion, and jump coefficients are in time-variable and position-parameters over a domain (e.g., a hyperbox or a manifold). A solution to the system is defined by a 4-tuple random vector-field process evolving in time. Since our unified system is a general-dimensional vector one with general nonlinearity and general high-order, the popular computation (e.g., integration by parts) based proof method can not be applied. Thus, we develop an approach to prove the well-posedness of an adapted 4-tuple strong solution to the system in a topological space and under a sequence of generalized local linear growth and Lipschitz conditions. As the further investigation of our system, we formulate a non-zero-sum stochastic differential game (SDG) problem with general number of players. By a 4-tuple solution to the system, we get a Pareto optimal Nash equilibrium policy to the SDG. In addition, illustrative examples from quantum physics, statistical mechanics, and queueing networks are also presented.