A Sufficient Condition for Linear-Quadratic Stochastic Zero-Sum Differential Games for Markov Jump Systems

In this note, we consider the linear-quadratic stochastic zero-sum differential game (LQ-SZSDG) for the Markov jump system (MJS) driven by Brownian motion. Unlike previous work considered in the literature, the diffusion term of the MJS is dependent on the state and the control of both players, and the cost parameters need not be definite matrices. We obtain a sufficient condition under which a feedback saddle point for the LQ-SZSDG exists. We show that the corresponding feedback saddle point is linear in the state and can be characterized in terms of a set of coupled Riccati differential equations (CRDEs). We also discuss the solvability of the CRDEs and verify the solvability through numerical examples under the existence condition of the saddle point.