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Time-inconsistent reinsurance and investment optimization problem with delay under random risk aversion

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

TL;DR

This work tackles a time-inconsistent, delay-inclusive reinsurance–investment optimization problem under random risk aversion. It develops a verification framework based on the expected certainty equivalent and a four-function trio $U$, $Y^{\gamma}$, $H$ to characterize equilibrium strategies, proving admissibility and optimality via a delay-aware pseudo HJB approach. The authors obtain (semi-)analytical equilibrium strategies and value functions under the CEV market for exponential utility and under Black–Scholes for both exponential and power utilities, including discrete and single-risk-aversion cases, with detailed numerical illustrations. The study demonstrates that random risk aversion and wealth-delay jointly influence reinsurance and investment decisions, and highlights how delay parameters interact with market and insurer characteristics to shape optimal strategies; it suggests extensions to VaR-constrained settings and bilateral reinsurance–investment games.

Abstract

This paper considers a newly delayed reinsurance and investment optimization problem incorporating random risk aversion, in which an insurer pursues maximization of the expected certainty equivalent of her/his terminal wealth and the cumulative delayed information of the wealth over a period. Specially, the insurer's surplus dynamics are approximated using a drifted Brownian motion, while the financial market is described by the constant elasticity of variance (CEV) model. Moreover, the performance-linked capital flow feature is incorporated and the wealth process is formulated via a stochastic delay differential equation (SDDE). By adopting a game-theoretic approach, a verification theorem with rigorous proofs is established to capture the equilibrium reinsurance and investment strategy along with the equilibrium value function. Furthermore, analytical or semi-analytical equilibrium reinsurance and investment strategies, together with their equilibrium value functions, are obtained under the CEV model for the exponential utility and derived under the Black-Scholes model for both exponential and power utilities. Finally, several numerical experiments are conducted to analyze the behavioral characteristics of the freshly-derived equilibrium reinsurance and investment strategy.

Time-inconsistent reinsurance and investment optimization problem with delay under random risk aversion

TL;DR

This work tackles a time-inconsistent, delay-inclusive reinsurance–investment optimization problem under random risk aversion. It develops a verification framework based on the expected certainty equivalent and a four-function trio , , to characterize equilibrium strategies, proving admissibility and optimality via a delay-aware pseudo HJB approach. The authors obtain (semi-)analytical equilibrium strategies and value functions under the CEV market for exponential utility and under Black–Scholes for both exponential and power utilities, including discrete and single-risk-aversion cases, with detailed numerical illustrations. The study demonstrates that random risk aversion and wealth-delay jointly influence reinsurance and investment decisions, and highlights how delay parameters interact with market and insurer characteristics to shape optimal strategies; it suggests extensions to VaR-constrained settings and bilateral reinsurance–investment games.

Abstract

This paper considers a newly delayed reinsurance and investment optimization problem incorporating random risk aversion, in which an insurer pursues maximization of the expected certainty equivalent of her/his terminal wealth and the cumulative delayed information of the wealth over a period. Specially, the insurer's surplus dynamics are approximated using a drifted Brownian motion, while the financial market is described by the constant elasticity of variance (CEV) model. Moreover, the performance-linked capital flow feature is incorporated and the wealth process is formulated via a stochastic delay differential equation (SDDE). By adopting a game-theoretic approach, a verification theorem with rigorous proofs is established to capture the equilibrium reinsurance and investment strategy along with the equilibrium value function. Furthermore, analytical or semi-analytical equilibrium reinsurance and investment strategies, together with their equilibrium value functions, are obtained under the CEV model for the exponential utility and derived under the Black-Scholes model for both exponential and power utilities. Finally, several numerical experiments are conducted to analyze the behavioral characteristics of the freshly-derived equilibrium reinsurance and investment strategy.

Paper Structure

This paper contains 25 sections, 15 theorems, 152 equations, 39 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model setup
  3. Surplus process
  4. Financial market
  5. Wealth process with delay
  6. Problem formulation and verification theorem
  7. Equilibrium strategies under the CEV model
  8. Special case with $n$ possible risk aversions
  9. Special case with one possible risk aversion
  10. Equilibrium strategies under the geometric Brownian motion
  11. The exponential utility
  12. The power utility
  13. Special case with $n$ possible risk aversions
  14. Special case with one possible risk aversion
  15. Numerical illustrations
  16. ...and 10 more sections

Key Result

Lemma 3.1

If $X^{u}(\cdot)\in\mathbb{S}^{2}_{\mathcal{F}}(0,T;\mathbb{R})$, then $X^{u}(\cdot),M_{1}^{u}(\cdot),M_{2}^{u}(\cdot)\in\mathbb{S}^{1}_{\mathcal{F}}(0,T;\mathbb{R})$ and $X^{u}(\cdot),M_{1}^{u}(\cdot),M_{2}^{u}(\cdot)\in\mathbb{L}^{1}_{\mathcal{F}}(0,T;\mathbb{R})$.

Figures (39)

  • Figure : (a) The impact of $\alpha$ on $\hat{q}(0)$
  • Figure : (a) The impact of $\alpha$ on $\hat{\pi}(0)$
  • Figure : (a) The impact of $\alpha$ on $\hat{q}(0)$ in case (I)
  • Figure : (a) The impact of $\beta$ on $\hat{q}(0)$
  • Figure : (a) The impact of $\mu_{1}$ on $\hat{q}(0)$
  • ...and 34 more figures

Theorems & Definitions (37)

  • Definition 3.1
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Remark 4.1
  • ...and 27 more