Two-player zero-sum stochastic differential games with regime switching and corresponding Hamilton-Jacobi-Bellman-Isaacs' equations.

In this paper, we study the two-player zero-sum stochastic differential games with regime switching. The stochastic system is driven by Brownian motion and Markov chains. The cost functionals are defined by controlled backward stochastic differential equations (BSDEs). In this framework, we allow the control processes to depend on the information from the past. By making full use of the construction of Markov chains, we prove that the upper value function and lower value function are indeed deterministic, while the cost functionals are random. We establish the dynamic programming principle (DPP), and obtain that the lower and upper value functions are the unique viscosity solutions of the corresponding systems for partial differential equations (PDEs) of Hamilton-Jacobi-Bellman-Isaacs' (HJBI) type. [ABSTRACT FROM AUTHOR]