Path - Dependent Stochastic Optimal Control And Differential Strategy

In this thesis we study the stochastic optimal control and differential game problems for path-dependent stochastic systems under recursive path-dependent cost functionals, whose associated Bellman and Isaacs equations from dynamic programming principle are path-dependent fully nonlinear partial differential equations(PDEs) of second order. A novel notion of viscosity solutions is introduced by restricting the semi-jets on an a-Holder path space Cα for α∈(0,1/2), and such a viscosity solution to the path-dependent Bellman equation is expected to be the value functional in a fairely general situation.In Chapter2, we introduce stochastic calculus on continuous and Holder continuous path spaces. The definition of our new viscosity solution to a Path-dependent PDE is given.In Chapter3we discuss path-dependent stochastic control problem. We first prove a general dynamic programming principle of the optimal control problem. Using Dupire’s functional Ito calculus and the technique of perturbation of the path, we prove that the value functional of the optimal stochastic control problem is a the viscosity solution to the associated path-dependent Bellman equation. A verification theorem and a state ap-proximation of the value functional are also given.In Chapter4, we discuss path-dependent stochastic differential game. We first prove a general dynamic programming principle of the game. Using the state approximating technique, we prove the existence of the game value under the Isaacs condition. We show that the value functionals of the stochastic differential game are the viscosity solutions to the associated path-dependent Isaacs equations. At last, we study the existence of ε-strategy for the stochastic differential game.