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Linear Quadratic Stochastic Two-Person Nonzero-Sum Differential Games: Open-Loop and Closed-Loop Nash Equilibria

In this paper, we consider a linear quadratic stochastic two-person nonzero-sum differential game. Open-loop and closed-loop Nash equilibria are introduced. The existence of the former is characterized by the solvability of a system of forward-backward stochastic differential equations, and that of the latter is characterized by the solvability of a system of coupled symmetric Riccati differential equations. Sometimes, open-loop Nash equilibria admit a closed-loop representation, via the solution to a system of non-symmetric Riccati equations, which is different from the outcome of the closed-loop Nash equilibria in general. However, it is found that for the case of zero-sum differential games, the Riccati equation system for the closed-loop representation of open-loop saddle points coincides with that for the closed-loop saddle points, which leads to the conclusion that the closed-loop representation of open-loop saddle points is the outcome of the corresponding closed-loop saddle point as long as both exist. In particular, for linear quadratic optimal control problem, the closed-loop representation of open-loop optimal controls coincides with the outcome of the corresponding closed-loop optimal strategy, provided both exist.