THE EXISTENCE OF GAME VALUE 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 under a recursive path-dependent cost functional. 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, and could be degenerate The dimension of the state space is high. The existence of approximating Nash equilibrium (epsilon-saddle point) is given under the formula about nonanticipative strategy with delay.