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

ArXiv ID: 2309.05003 “View on arXiv”

Authors: Unknown

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.

Keywords: stochastic linear-quadratic differential games, backward stochastic differential equation, portfolio selection, regime switching, short-selling constraints

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper introduces a novel and highly advanced mathematical framework involving multidimensional indefinite stochastic Riccati equations and BSDEs with non-Markovian regime switching, representing a frontier in stochastic control theory. However, it contains no backtesting, datasets, or implementation details, relying solely on theoretical proofs and a portfolio selection example without empirical validation.

  flowchart TD
    A["Research Goal<br>Zero-sum stochastic LQ games<br>with non-Markovian regime switching"] --> B{"Data / Inputs"}
    B --> B1["Stochastic Differential Equations<br>with regime switching"]
    B --> B2["Zero-sum payoff structure"]
    B --> B3["Control Constraints<br>(e.g., No-shorting)"]
    B --> B4["Random coefficients"]
    B --> B5["Unbounded coefficients"]
    A --> C["Methodology<br>Multidimensional Stochastic Riccati Equation<br>+<br>Multidimensional BSDE"]
    C --> D{"Computational Process"}
    D --> D1["Solve Indefinite Stochastic Riccati Equation"]
    D --> D2["Solve Multidimensional BSDE"]
    D1 --> D3["Prove Existence & Uniqueness<br>Solution can be positive & negative"]
    D2 --> D3
    D3 --> E["Key Outcomes"]
    E --> E1["Closed-loop optimal feedback<br>control-strategy pairs"]
    E --> E2["Application: Portfolio selection games<br>with no-shorting constraints"]