Open-loop and closed-loop solvabilities for zero-sum stochastic linear quadratic differential games of Markovian regime switching system

This paper investigates zero-sum stochastic linear quadratic (SLQ) differential games with Markovian jumps. Open-loop and closed-loop solvabilities are studied by employing a new ``decomposition method", which decomposes the open-loop and closed-loop solvability problems of zero-sum SLQ differential games into two coupled SLQ control problems. Under the uniform convexity-concavity condition, we construct the open-loop saddle point along with its closed-loop representation based on the solution to a system of constrained coupled differential Riccati equations (CDREs), whose solvability is also established by employing the dimension extension technique and the continuation method. Finally, we provide a concrete example and present its closed-form saddle point based on the theoretical results obtained.