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This paper is concerned with zero-sum stochastic linear-quadraticdifferential games in a regime switching model. The coefficients of the gamesdepend on the underlying noises, so it is a non-Markovian regime switchingmodel. Based on the solutions of a new kind of multidimensional indefinitestochastic Riccati equation (SRE) and a multidimensional linear backwardstochastic differential equation (BSDE) with unbounded coefficients, we provideclosed-loop optimal feedback control-strategy pairs for the two players. Themain contribution of this paper, which is of great importance in its own rightfrom the BSDE theory point of view, is to prove the existence and uniqueness ofthe solution to the new kind of SRE. Notably, the first component of thesolution (as a process) is capable of taking positive and negative valuessimultaneously. For homogeneous systems, we obtain the optimal feedbackcontrol-strategy pairs under general closed convex cone control constraints.Finally, these results are applied to portfolio selection games with full orpartial no-shorting constraint in a regime switching market with randomcoefficients.

Multidimensional indefinite stochastic Riccati equations and zero-sum  stochastic linear-quadratic differential games with non-Markovian regime  switching