A Feedback Nash Equilibrium for Affine-Quadratic Zero-Sum Stochastic Differential Games with Random Coefficients

We study the affine-quadratic zero-sum stochastic differential game with random coefficients, where the coefficients of the stochastic differential equation (SDE) are random processes and both additive and state multiplicative noise are included in the diffusion term of the corresponding SDE. By applying Ito-Kunita’s formula to the quadratic random field, we develop a direct approach, also known as the completion of squares method, to characterize the explicit (feedback) Nash equilibrium and obtain the optimal game value. The characterized Nash equilibrium depends linearly on the state and the additional linear backward SDE. We also verify the optimality of the Nash equilibrium by characterizing the smooth solution of the stochastic Hamilton-Jacobi-Isaacs equation that is the second-order stochastic partial differential equation obtained from dynamic programming.