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This paper is concerned with two-player zero-sum linear-quadratic stochasticdifferential games in a regime switching model. The controlled inhomogeneoussystem coefficients depend on the underlying noises, so it is a non-Markovianregime switching model. Based on a new kind of multidimensional indefinitestochastic Riccati equation (SRE) and a multidimensional linear backwardstochastic differential equation (BSDE) with unbounded coefficients, we canprovide optimal feedback control-strategy pairs for the two players in aclosed-loop form. The main contribution of this paper, which is important inits own right from the BSDE theory point of view, is to prove the existence anduniqueness of the new kind of multidimensional indefinite SRE. Interestingly,the components of the solution can take positive, zero and negative valuessimultaneously. We also obtain the corresponding optimal feedbackcontrol-strategy pairs for homogeneous systems under closed convex cone controlconstraints. Finally, these results are applied to portfolio selection problemswith different short-selling prohibition constraints in a regime switchingmarket with random coefficients.

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