Multidimensional Indefinite Stochastic Riccati Equations and Zero-Sum Stochastic Linear-Quadratic Differential Games with Non-Markovian Regime Switching

This paper is concerned with zero-sum stochastic linear-quadratic differential games in a regime-switching model. The coefficients of the games depend on the underlying noises, so it is a non-Markovian regime-switching model. Based on the solutions of a new kind of multidimensional indefinite stochastic Riccati equation (SRE) and a multidimensional linear backward stochastic differential equation (BSDE) with unbounded coefficients, we provide closed-loop optimal feedback control-strategy pairs for the two players. The main contribution of this paper, which is of great importance in its own right from the BSDE theory point of view, is to prove the existence and uniqueness of the solution to the new kind of SRE. Notably, the first component of the solution (as a process) is capable of taking positive and negative values simultaneously. For homogeneous systems, we obtain the optimal feedback control-strategy pairs under general closed convex cone control constraints. Finally, these results are applied to portfolio selection games with full or partial no-shorting constraint in a regime-switching market with random coefficients.