Necessary and sufficient conditions for Pareto optimality of the stochastic systems in finite horizon

This paper is concerned with the necessary and sufficient conditions for the Pareto optimality in the finite horizon stochastic cooperative differential game. Based on the necessary and sufficient characterization of the Pareto optimality, the problem is transformed into a set of constrained stochastic optimal control problems with a special structure. Utilizing the stochastic Pontryagin minimum principle, the necessary conditions for the existence of the Pareto solutions are put forward. Under certain convex assumptions, it is shown that the necessary conditions are also sufficient ones. Next, we study the indefinite linear quadratic (LQ) case. It is pointed out that the solvability of the related generalized differential Riccati equation (GDRE) provides the sufficient condition under which all Pareto efficient strategies can be obtained by the weighted sum optimality method. Two examples shed light on the effectiveness of theoretical results.