Linear Quadratic Nonzero-Sum Mean-Field Stochastic Differential Games with Regime Switching

Abstract

This paper is concerned with a linear quadratic (LQ) nonzero-sum stochastic differential game for regime switching diffusions with mean-field interactions. The salient features of this paper include that the concept of strategies is first adopted in the LQ nonzero-sum game and conditional mean-field terms appear in the state equation and cost functionals. First, a candidate optimal feedback control-strategy pair for the two players is formally constructed based on solutions of four coupled Riccati equations. Then, we verify that the formal optimal pair is indeed a Nash equilibrium for the game by a delicate multi-step completion of squares. The four Riccati equations introduced in this paper are new in the literature. Uniqueness of solutions to the Riccati equations for the general case and existence of solutions for a special case are obtained. Finally, a numerical example is reported to demonstrate the theoretical results.