Some Linear-Quadratic Stochastic Differential Games Driven by State Dependent Gauss-Volterra Processes

Abstract

In this paper a two player zero sum stochastic differential game in a finite dimensional space having a linear stochastic equation with a state dependent Gauss-Volterra noise is formulated and solved with a quadratic payoff for the two players and a finite time horizon. The control strategies are continuous linear state feedbacks. A Nash equilibrium is verified for the game and the two optimal strategies are obtained using a direct method that does not require solving nonlinear partial differential equations or forward-backward stochastic differential equations. The Gauss-Volterra processes are singular integrals of a standard Brownian motion and include various types of fractional Brownian motions as well as some other Gaussian processes.