Linear-Quadratic Two-Person Differential Games

The purpose of this chapter is to develop a theory for stochastic linear-quadratic two-person differential games. Open-loop and closed-loop Nash equilibria are explored in the context of nonzero-sum and zero-sum differential games. The existence of an open-loop Nash equilibrium is characterized in terms of a system of constrained forward-backward stochastic differential equations, and the existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of coupled symmetric differential Riccati equations. It is shown that in the nonzero-sum case, the closed-loop representation for open-loop Nash equilibria is different from the outcome of closed-loop Nash equilibria in general, whereas they coincide in the zero-sum case when both exist. Some results for infinite-horizon zero-sum differential games are also established in terms of algebraic Riccati equation.