A Variational Formula for Nonzero-Sum Stochastic Differential Games of FBSDEs and Applications

A nonzero-sum stochastic differential game problem is investigated for fully coupled forward-backward stochastic differential equations (FBSDEs in short) where the control domain is not necessarily convex. A variational formula for the cost functional in a given spike perturbation direction of control processes is derived by the Hamiltonian and associated adjoint systems. As an application, a global stochastic maximum principle of Pontryagin’s type for open-loop Nash equilibrium points is established. Finally, an example of a linear quadratic nonzero-sum game problem is presented to illustrate that the theories may have interesting practical applications and the corresponding Nash equilibrium point is characterized by the optimality system. Here the optimality system is a fully coupled FBSDE with double dimensions (DFBSDEs in short) which consists of the state equation, the adjoint equation, and the optimality conditions.