Robust mean-variance stochastic differential reinsurance and investment games under volatility risk and model uncertainty

Paper: arXiv 2412.09171

Authors: Guohui Guan, Zongxia Liang, Yi Xia

Abstract

This paper investigates robust stochastic differential games among insurers under model uncertainty and stochastic volatility. The surplus processes of ambiguity-averse insurers (AAIs) are characterized by drifted Brownian motion with both common and idiosyncratic insurance risks. To mitigate these risks, AAIs can purchase proportional reinsurance. Besides, AAIs allocate their wealth in a financial market consisting of cash, and a stock characterized by the 4/2 stochastic volatility model. AAIs compete with each other based on relative performance with the mean-variance criterion under the worst-case scenario. This paper formulates a robust time-consistent mean-field game in a non-linear system. The AAIs seek robust, time-consistent response strategies to achieve Nash equilibrium strategies in the game. We introduce $n$-dimensional extended Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations and corresponding verification theorems under compatible conditions. Semi-closed forms of the robust $n$-insurer equilibrium and mean-field equilibrium are derived, relying on coupled Riccati equations. Suitable conditions are presented to ensure the existence and uniqueness of the coupled Riccati equation as well as the integrability in the verification theorem. As the number of AAIs increases, the results in the $n$-insurer game converge to those in the mean-field game. Numerical examples are provided to illustrate economic behaviors in the games, highlighting the herd effect of competition on the AAIs.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical exploration of stochastic differential games using advanced mathematical tools such as $n$-dimensional extended HJBI equations, coupled Riccati equations, and the 4/2 stochastic volatility model. However, it relies entirely on synthetic numerical examples rather than real-world data validation or empirical backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive robust time-consistent reinsurance & investment strategies under stochastic volatility and model uncertainty] --> B[Inputs: 4/2 Stochastic Volatility, Drifted Brownian Surplus, Relative Performance, Ambiguity Aversion]
    B --> C[Formulate Game: Set up Robust n-Insurer & Mean-Field Games under Worst-Case Scenarios]
    C --> D[Solve HJBI System: Construct n-Dimensional Extended HJBI Equations & Verification Theorems]
    D --> E[Derive Semi-Closed Form Solutions via System of Coupled Riccati Equations]
    E --> F[Outcomes: Prove Convergence to Mean-Field Equilibrium & Demonstrate Competition Herd Effect via Numerical Examples]

Cited References (117)

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6. COVID -19, volatility dynamics, and sentiment trading (2021)
7. Robust optimal investment and reinsurance problem for a general insurance company under H eston model (2017)
8. Robust reinsurance and investment strategies under principal–agent framework (2022)

0 of 117 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.