  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.
