Infinite Horizon Linear-Quadratic Leader-Follower Stochastic Differential Games for Regime Switching Diffusions

Abstract

This paper studies a discounted linear-quadratic (LQ) leader-follower stochastic differential game for regime switching diffusion in an infinite horizon. Within the \(L^{2,r}\)-stabilizability framework, we first, as a preliminary, establish the global well-posedness of infinite horizon linear stochastic differential equations and backward stochastic differential equations with Markov chains. Next, under the uniform convexity condition for LQ problems, we obtain an open-loop Stackelberg equilibrium of the leader-follower game. By employing the so-called four-step scheme, the corresponding Hamiltonian systems for the two players are decoupled and then the open-loop Stackelberg equilibrium admits a state feedback representation in terms of two new-type algebraic Riccati equations together with some certain stabilizing condition. Finally, we report a numerical example to illustrate our theoretical results, including the solutions to the Riccati equations, the Stackelberg equilibrium strategies, and the behavior of the corresponding state process.