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

TL;DR

This paper develops a novel multidimensional indefinite stochastic Riccati equation framework to solve non-Markovian regime switching zero-sum stochastic linear-quadratic differential games, providing explicit feedback strategies.

Contribution

It introduces a new class of multidimensional indefinite stochastic Riccati equations and proves their existence and uniqueness, enabling solutions to complex non-Markovian differential games.

Findings

Established existence and uniqueness of solutions to the new SREs.

Derived explicit feedback control strategies for the differential games.

Applied results to portfolio selection with regime switching and constraints.

Abstract

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.