Abstract
AbstractThis chapter is concerned with necessary/sufficient conditions for Pareto optimality in finite horizon cooperative stochastic differential games. Based on the equivalent characterization of Pareto optimality, the problem is transformed into a set of constrained stochastic optimal control problems with a special structure. Employing the stochastic Pontryagin’s minimum principle, necessary conditions for the existence of Pareto-efficient strategies are put forward. Under certain convex assumptions, it is shown that the necessary conditions are also the sufficient ones. Next, the obtained results are extended to the indefinite LQ case. Necessary conditions deriving from the minimum principle as well as convexity condition on the cost functional provide the sufficient conditions for a open-loop control to be Pareto efficient. In addition, the solvability of the related stochastic differential Riccati equation (SDRE) provides the sufficient condition under which all closed-loop Pareto-efficient strategies can be obtained by the weighted sum optimality method.
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