Existence of Game Value and Approximating Nash Equilibrium for Path-dependent Stochastic Differential Game

摘要

In this paper we study a two-player zero-sum stochastic differential game for a path-dependent stochastic system. Two basic theoretic problems in SDG are addressed: existence of game value and Nash equilibrium. Due to the typical non-Markovian structure, the game value is a random field. Dividing the time horizontal into small intervals, we approximate the path-dependent game by a series of state-dependent games. We utilize the state-dependent viscosity solution theory to prove that, under Isaacs' condition the game value exists. In our model, coefficients of diffusion of the system contain control and strategy, are path-dependent, and could be degenerate. The dimension of the state space is high. The existence of approximating Nash equilibrium is given under the formula about nonanticipative strategy with delay.