Linear Exponential Quadratic Stochastic Differential Games

In this technical note a two person noncooperative stochastic differential game described by a linear stochastic equation with a Brownian motion and a payoff that is the exponential of a quadratic functional in the state and the control strategies of the two players is explicitly solved. The optimal strategies form a Nash equilibrium. The value of the game is also determined explicitly. The solution of this problem is naturally related to a linear quadratic stochastic differential game and a linear exponential quadratic control problem.