Mathematical Sciences: Research on Optimal Stochastic Control and Related Topics

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This project includes several subprojects and represents quite a vast research program in the area of stochastic control, stochastic processes and nonlinear filtering. The research in stochastic control proposed by Professor Fleming is focused on the questions related to the existence and properties of solutions to the Hamilton-Jacobi equation. One outstanding question is whether given the Hamiltonian function there is a corresponding stochastic control problem for diffusion processes leading to this function. Recently, Fleming and Souganidis introduced a new formulation of a stochastic differential game, based on a new deterministic approach to differential games, which leads to the solution of this problem by replacing the control problem by a game problem. Fleming also intends to work on a number of open problems related to viscosity solutions for nonlinear first- order partial differential equations. Among open problems to be considered are the theory of discontinuous viscosity solutions, and the application of duality theorems of convex analysis to approximate the value function of the Hamilton-Jacobi equation. The research in stochastic processes centers on the exit problem of Freidlin-Wentsel theory. Fleming introduced the stochastic control approach to this problem and applied an asymptotic expansion to the dynamic programming equation obtained after a logarythmic transformation. This enables him to compute exit probabilities from the expansion. These expansions are of interest for problems of rare overloads in communications channels. On the nonlinear filtering problem, Fleming proposes to use some asymptotic techniques to describe analytically the structure of the nonlinear filter for small intensities of noise in the output equation. Stochastic control is a branch of applied mathematics and engineering system theory that deals with control of dynamical systems in situations involving random inputs and measurement noise. The mathematical tools involved in this research are the theory of stochastic processes, nonlinear partial differential equations, asymptotic methods and numerical methods.