Linear Quadratic Two-Person Zero-Sum Stochastic Differential Games

In this paper, we consider a linear quadratic stochastic two-person zero-sum dif-ferential game (SLQG, for short). The main purpose is to characterize both open-loop and closed-loop saddle points of the game. In our framework, the state equation is non-homogeneous; the controls for both players are allowed to appear in both drift and diffusion of the state equation; the weighting matrices in the performance functional are not assumed to be definite/non-singular; the cross-terms between the state process and controls and the first-order terms are allowed to appear. Both finite and infinite horizon cases are discussed in this paper.For the game in a finite horizon, the coefficients of the state equation are allowed to be unbounded, and a terminal penalty term is allowed to appear in the performance functional. The existence and uniqueness of solutions to the state equation are proved under certain conditions. By a variational method, the existence of an open-loop sad-dle point is characterized by the existence of an adapted solution to a linear forward-backward stochastic differential equation with constraints, together with a convexity-concavity condition. We further discuss the relation between open-loop saddle points and open-loop lower/upper value functions. Zhang [SIAM J. Control Optim.,2005,43:2157-2165] proved that for a linear quadratic deterministic two-person zero-sum dif-ferential game, the the existence of an open-loop saddle point is equivalent to the finite-ness of the corresponding open-loop lower and upper values, which is also equivalent to the existence of the open-loop value. However, such a result does not hold in general. Instead, we obtain a general weaker conclusion:The finiteness of the open-loop lower and upper values implies the convexity-concavity condition mentioned above. Using a decoupling technique, a Riccati differential equation is deduced, and the existence of a closed-loop saddle point is characterized by the existence of a regular solution to the corresponding Riccati differential equation. Also, the representations of the closed-loop saddle points and the value function are given. We indicate that the solution of the Riccati equation may be non-unique, while the regular solution must be unique if it exists. Comparing the characterizations of open-loop and closed-loop saddle points, we find that the convexity-concavity condition for the performance functional is neces-sary for the existence of an open-loop saddle point, but not necessary for the existence of a closed-loop saddle point. Therefore, the existence of a closed-loop saddle point does not imply that of an open-loop saddle point. However, the existence of a closed-loop saddle point does imply the solvability of the linear forward-backward stochastic differential equation with constraints. On the other hand, because of the regularity re-quirement of the solution to the Riccati equation, the existence of an open-loop saddle point does not imply that of a closed-loop saddle point either. Also, it is found that there is an essential difference between linear quadratic stochastic optimal control (SLQ, for short) and SLQG problems:For the SLQ problems, the existence of a closed-loop s-trategy implies the existence of an open-loop optimal control. However, the existence of a closed-loop saddle point does not necessarily imply the existence of an open-loop saddle point. Hence, SLQ problems can only remain a formal special case of SLQG problems.For the game in an infinite horizon, we assume that the coefficient matrices in the state equation and the weighting matrices in the performance functional are all constan-t. As a crucial step, it is proved that under some mild conditions, there exists a unique square-integrable solution/adapted solution to a class of linear forward/backward s-tochastic differential equations in an infinite horizon. When the coefficient matrices of the state equation satisfy certain condition, the existence of open-loop saddle points can be characterized in a similar way to the finite horizon case. The only difference is that the forward-backward stochastic differential equation is defined in an infinite horizon now, so we need to discuss the existence of L2-adapted solutions. To charac-terize closed-loop saddle points, we introduce the algebraic Riccati equation (ARE, for short) and its stabilizing solution. By a method of matrix inequality, the existence of a closed-loop saddle point is characterized by the existence of a stabilizing solution to the corresponding ARE. Further, we give the representations of the closed-loop saddle points and the value function.The paper begins with a brief review of the history and development of differential games, especially the SLQG problems. Then it discusses the well-posedness of linear forward/backward stochastic differential equations in a finite/infinite horizon. Next, it is devoted to the characterizations of open-loop and closed-loop saddle points of SLQG problems in a finite/infinite horizon, and some illustrative examples are given. Finally, some concluding remarks and related problems are collected in the last chapter.