Paper: arXiv 2508.15355

Authors: Liyuan Cui, Wenyuan Li

Abstract

This paper investigates optimal investment and insurance strategies under a mean-variance criterion with path-dependent effects. We use a rough volatility model and a Hawkes process with a power kernel to capture the path dependence of the market. By adding auxiliary state variables, we degenerate a non-Markovian problem to a Markovian problem. Next, an explicit solution is derived for a path-dependent extended Hamilton-Jacobi-Bellman (HJB) equation. Then, we derive the explicit solutions of the problem by extending the Functional Ito formula for fractional Brownian motion to the general path-dependent processes, which includes the Hawkes process. In addition, we use earthquake data from Sichuan Province, China, to estimate parameters for the Hawkes process. Our numerical results show that the individual becomes more risk-averse in trading when stock volatility is rough, while more risk-seeking when considering catastrophic shocks. Moreover, an individual’s demand for catastrophe insurance increases with path-dependent effects. Our findings indicate that ignoring the path-dependent effect would lead to a significant underinsurance phenomenon and highlight the importance of the path-dependent effect in the catastrophe insurance pricing.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous theoretical framework extending functional Itô calculus to a mixed fractional Brownian-Hawkes process with power kernels for optimal investment and insurance problems. The empirical work uses real Sichuan earthquake data for calibration, successfully bridging complex non-Markovian stochastic control theory with practical actuarial insights on underinsurance.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimal Investment & Catastrophe Insurance Strategy under Path Dependence] --> B[Model Setup: Rough Volatility & Hawkes Process with Power Kernel]
    C[Data Input: Sichuan Earthquake Data for Hawkes Parameter Estimation] --> D
    B --> D[Methodology: Degenerate to Markovian via Auxiliary Variables]
    D --> E[Derivation: Extend Functional Itô Formula & Solve Path-Dependent Extended HJB Equation]
    E --> F[Key Outcomes: <br>1. Rough volatility increases trading risk aversion<br>2. Catastrophic shocks increase risk-seeking behavior<br>3. Path dependence boosts insurance demand; ignoring it leads to severe underinsurance]

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