Stochastic Differential Equations and Applications

Purdue University, West Lafayette IN

Investigators

Abstract

0204332 Ma The principal investigator proposes to study various issues involving stochastic differential equations (hereafter SDEs), stochastic partial differential equations (SPDEs), and their relations with partial differential equations (PDEs). A new "functional" form of nonlinear Feynman-Kac formula is sought, via a study of the general theory of backward stochastic differential equations (BSDEs) with non-Lipschitz coefficients. The path regularity of the solutions to reflected backward SDEs (RBSDEs) will be studied, with an eye on its application to the numerical method for such equations. The study will also lead to the regularity of solutions to a class of obstacle problems in PDEs, via some new probabilistic representation formulae for the derivatives of such solutions. The PI proposes to continue his research on the new notion of stochastic viscosity solution for nonlinear SPDEs, including the stability and uniqueness results in the fully nonlinear case. Some stochastic control/stochastic finance problems, inspired by a risk reserve model involving investment, will receive strong attention. One example is the optimal retention/investment problem for an insurance company that uses the ruin probability as its stability criterion. The Feynman-Kac Formula is a powerful tool that links probability theory to analysis, especially to the theory of partial differential equations. The nonlinear form of this formula is useful for studying many nonlinear parabolic partial differential equations such as reaction diffusion equations, derived from expansion of advantaged genes in biology, combustion theory, as well as chemical kinetics. The celebrated Black-Scholes formula in modern stochastic finance theory and the Hamilton-Jacobi-Bellman equation in stochastic control theory are also, in essence, examples of the formula. The main part of the proposed research is to discover the possibility of advancing such formulae to a higher level, either by considering the "functional form", or by considering the "stochastic form" (viscosity solution of fully nonlinear SPDEs). Some stochastic control problems arising from finance, insurance, and many other fields are naturally considered as applications of the theory.