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Existence Conditions of Pareto Solutions in Finite Horizon Stochastic Differential Games

In: Essays on Pareto Optimality in Cooperative Games

Author

Listed:
  • Yaning Lin

    (Shandong University of Technology, School of Mathematics and Statistics)

  • Weihai Zhang

    (Shandong University of Science and Technology, College of Electrical Engineering and Automation)

Abstract

This 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.

Suggested Citation

  • Yaning Lin & Weihai Zhang, 2022. "Existence Conditions of Pareto Solutions in Finite Horizon Stochastic Differential Games," Springer Books, in: Essays on Pareto Optimality in Cooperative Games, chapter 0, pages 7-30, Springer.
    • Handle: RePEc:spr:sprchp:978-981-19-5049-0_2
    DOI: 10.1007/978-981-19-5049-0_2
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