Zero sum two-player differential game under three regimes

This paper investigates a two player zero-sum stochastic differential game characterized by three distinct regimes, reflecting the regime switching dynamics within the system. We aim to derive an explicit solution for a number of configurations of the switching system by means of the viscosity solutions approach. In particular, we analyze the associated Hamilton-Jacobi-Bellman-Isaacs equations and examine how the interactions between the different regimes affect the structure of the value function. The proposed framework provides a systematic approach to characterize the optimal strategies of the two players and to obtain explicit representations of the value function un der suitable assumptions. We also illustrate how to derive the value functions in case we know the qualitative structure of switching regions.