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Equilibrium reinsurance and investment strategies for insurers with random risk aversion under Heston's SV model

Jian-hao Kang, Zhun Gou, Nan-jing Huang

TL;DR

This work addresses dynamic reinsurance and investment for an insurer with random risk aversion in a market where the risky asset follows Heston's stochastic volatility. By formulating a time-inconsistent problem through expected certainty equivalents and applying a game-theoretic equilibrium with a verification theorem, the authors derive (semi-)analytic equilibrium strategies under exponential utility, including both an $n$-point and a one-point risk-aversion setting. Key contributions include a rigorous verification framework, explicit equilibrium controls that are deterministic and state-independent, and numerical analyses illustrating how model parameters shape reinsurance and investment decisions. The results offer insight into how random risk preferences interact with stochastic volatility to drive lifecycle investment and risk-transfer choices, with implications for insurer risk management under realistic market dynamics.

Abstract

This study employs expected certainty equivalents to explore the reinsurance and investment issue pertaining to an insurer that aims to maximize the expected utility while being subject to random risk aversion. The insurer's surplus process is modeled approximately by a drifted Brownian motion, and the financial market is comprised of a risk-free asset and a risky asset with its price depicted by Heston's stochastic volatility (SV) model. Within a game theory framework, a strict verification theorem is formulated to delineate the equilibrium reinsurance and investment strategies as well as the corresponding value function. Furthermore, through solving the pseudo Hamilton-Jacobi-Bellman (HJB) system, semi-analytical formulations for the equilibrium reinsurance and investment strategies and the associated value function are obtained under the exponential utility. Additionally, several numerical experiments are carried out to demonstrate the characteristics of the equilibrium reinsurance and investment strategies.

Equilibrium reinsurance and investment strategies for insurers with random risk aversion under Heston's SV model

TL;DR

This work addresses dynamic reinsurance and investment for an insurer with random risk aversion in a market where the risky asset follows Heston's stochastic volatility. By formulating a time-inconsistent problem through expected certainty equivalents and applying a game-theoretic equilibrium with a verification theorem, the authors derive (semi-)analytic equilibrium strategies under exponential utility, including both an -point and a one-point risk-aversion setting. Key contributions include a rigorous verification framework, explicit equilibrium controls that are deterministic and state-independent, and numerical analyses illustrating how model parameters shape reinsurance and investment decisions. The results offer insight into how random risk preferences interact with stochastic volatility to drive lifecycle investment and risk-transfer choices, with implications for insurer risk management under realistic market dynamics.

Abstract

This study employs expected certainty equivalents to explore the reinsurance and investment issue pertaining to an insurer that aims to maximize the expected utility while being subject to random risk aversion. The insurer's surplus process is modeled approximately by a drifted Brownian motion, and the financial market is comprised of a risk-free asset and a risky asset with its price depicted by Heston's stochastic volatility (SV) model. Within a game theory framework, a strict verification theorem is formulated to delineate the equilibrium reinsurance and investment strategies as well as the corresponding value function. Furthermore, through solving the pseudo Hamilton-Jacobi-Bellman (HJB) system, semi-analytical formulations for the equilibrium reinsurance and investment strategies and the associated value function are obtained under the exponential utility. Additionally, several numerical experiments are carried out to demonstrate the characteristics of the equilibrium reinsurance and investment strategies.

Paper Structure

This paper contains 15 sections, 4 theorems, 91 equations, 24 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model setting and assumptions
  3. Surplus process
  4. Financial market
  5. Wealth process
  6. Problem formulation and verification theorem
  7. Equilibrium strategies to the optimization problem
  8. Special case with $n$ possible risk aversions
  9. Special case with one possible risk aversion
  10. Numerical illustrations
  11. Impacts of model parameters on the equilibrium investment strategy
  12. Impacts of model parameters on the equilibrium reinsurance strategy
  13. Conclusions
  14. Proof of Theorem \ref{['theorem3.1']}
  15. Proof of Proposition \ref{['proposition4.1']}

Key Result

Theorem 3.1

(Verification theorem) Assume that there exist functions $U, Y^{\gamma}, H\in C^{1,2,2}(\mathcal{Q})$ such that and with boundary conditions $U(T,x,v)=x$ and $Y^{\gamma}(T,x,v)=\varphi^{\gamma}(x)$ for all $\gamma$, where and Moreover, assume that $U$, $Y^{\gamma}$ and $H$ satisfy the condition (A) for all $\gamma$. Then $\hat{u}=(\hat{q},\hat{\pi})$ is an equilibrium reinsurance and investmen

Figures (24)

  • Figure : (d) Case (II) when $T=100$
  • Figure : (d) Case (II) when $T=100$
  • Figure : (d) Case (II) when $T=100$
  • Figure : (d) Case (II) when $T=100$
  • Figure : (d) Case (II) when $T=100$
  • ...and 19 more figures

Theorems & Definitions (15)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • Remark 4.1
  • Proposition 4.1
  • proof
  • Remark 4.2
  • ...and 5 more