Closed-Loop Strategies in Stochastic Linear-Quadratic Problems

报告题目:Closed-Loop Strategies in Stochastic Linear-Quadratic Problems




报告摘要:In stochastic differential games, open-loop controls depend on the initial state as well as all the information, including those of the opponent, over the whole time duration. This makes open-loop controls practically infeasible. In this talk, we introduce closed-loop strategies for stochastic linear-quadratic (LQ) problems, which are non-anticipating and hence more meaningful and convenient to use in reality. We begin with LQ control problems, carefully illustrate the relationship between open-loop and closed-loop solvabilities, and provide characterizations of the existence of open-loop optimal controls and closed-loop optimal strategies. Then we consider stochastic LQ two-person differential games. Besides characterizing open-loop and closed-loop Nash equilibria, we find several interesting but unexpected facts: 1) The existence of an open-loop Nash equilibrium is not equivalent to that of a closed-loop Nash equilibrium. 2) The outcome of a closed-loop Nash equilibrium is different from the closed-loop representation of an open-loop Nash equilibrium. 3) For the case of zero-sum differential games, the closed-loop representation of open-loop saddle points coincides with the outcome of the corresponding closed-loop saddle point as long as both exist.