Linear Quadratic Pareto Game of the Stochastic Systems in Infinite Horizon

Abstract

This paper investigates the necessary/sufficient conditions for Pareto optimality in the infinite horizon linear quadratic stochastic 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. Under the assumption about the Lagrange multipliers, utilizing the stochastic Pontryagin maximum principle, the necessary conditions for the existence of the Pareto efficient strategies are presented. Furthermore, a condition is introduced to guarantee that the element zero does not belong to the Lagrange multiplier set. In addition, the necessary conditions, the convexity condition on the weighted sum cost functional and a transversality condition provide the sufficient conditions for a control to be Pareto efficient. The characterization of Pareto efficient strategies and Pareto solutions is also studied. If the system is stabilizable, then the solvability of the related generalized algebraic Riccati equation provides a sufficient condition under which all Pareto efficient strategies can be obtained by the weighted sum optimality method and all Pareto solutions can be derived based on the solutions of an introduced algebraic Lyapunov equation.