Optimal reinsurance-investment problem for a general insurance company under a generalized dynamic contagion claim model

Fan Wu; Xin Zhang; Zhibin Liang

doi:10.3934/mcrf.2022030

Article Contents

2023, Volume 13, Issue 3: 1131-1159. Doi: 10.3934/mcrf.2022030

This issue Previous Article Control theory approach to continuous-time finite state mean field games Next Article Feedback law to stabilize linear infinite-dimensional systems

Optimal reinsurance-investment problem for a general insurance company under a generalized dynamic contagion claim model

1.
School of Mathematics, Southeast University, Nanjing 211189, China
2.
School of Mathematical Science, Nanjing Normal University, Nanjing 210023, China

^*Corresponding author: Xin Zhang
^*Corresponding author: Xin Zhang

Received: October 2021

Revised: May 2022

Early access: July 2022

Published: September 2023

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Abstract

In this paper, we study an optimal management problem for a general insurance company which holds shares of an insurance company and a reinsurance company. The general company aims to derive the equilibrium reinsurance-investment strategy under the mean-variance criterion. The claim process described by a generalized compound dynamic contagion process introduced by [18] which allows for self-exciting and externally-exciting clustering effect for the claim arrivals and the processes of the risky assets are described by the jump-diffusion models. Based on practical considerations, we suppose that the externally-exciting clustering effect will simultaneously affect both the price of risky assets and the intensity of claims. To overcome the inconsistency issue caused by the mean-variance criterion, we formulate the optimization problem as an embedded game and solve it via a corresponding extended Hamilton-Jacobi-Bellman equation. The equilibrium reinsurance-investment strategy is obtained, which depends on a solution to an ordinary differential equation. In addition, we demonstrate the derived equilibrium strategy and the economic implications behind it through a large number of mathematical analysis and numerical examples.

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Mathematics Subject Classification: Primary: 49L20, 49N90; Secondary: 91G80.

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Figure 1. The equilibrium reinsurance-investment strategies and value function

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Figure 2. Impact of parameters $ \alpha $ and $ m_{R} $ on the function $ h(t) $

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