Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market

Sheng Li; Wei Yuan; Peimin Chen

doi:10.3934/jimo.2022068

Article Contents

2023, Volume 19, Issue 4: 2855-2888. Doi: 10.3934/jimo.2022068

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Optimal control on investment and reinsurance strategies with delay and common shock dependence in a jump-diffusion financial market

1.
School of Statistics, Chengdu University of Information Technology, Chengdu, 610103, China
2.
Sichuan Administration Institute, Chengdu, 610072, China
3.
School of hospitality Management, Shanghai Business School, Shanghai, 200235, China

^*Corresponding author: Peimin Chen
^*Corresponding author: Peimin Chen

Received: February 2021

Revised: February 2022

Published: April 2023

The Sheng Li is supported by the project of the Chengdu University of Information Technology Introducing Talents to Launch Scientific Research (KYTZ202193)

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Abstract

In this paper, we consider an optimal mean-variance investment and reinsurance problem with delay and Common Shock Dependence. An insurer can control the claim risk by purchasing proportional reinsurance. He/she invests his/her wealth on a risk-free asset and a risky asset, which follows the jump-diffusion process. By introducing a capital flow related to the historical performance of the insurer, the wealth process described by a stochastic differential equation with delay is obtained. By stochastic linear-quadratic control theory and stochastic control theory with delay, we achieve the explicit expression of the optimal strategy and value function in the framework of the viscosity solution. Furthermore, an efficient strategy and its efficient frontier are derived by Lagrange dual method. Finally, we analyze the influence of the parameters of our model on the efficient frontier by a numerical example.

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Mathematics Subject Classification: Primary: 91B30, 93E20; Secondary: 62P05.

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Figure 1. The efficient frontiers for different $ A $

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Figure 2. The efficient frontiers for different $ h $

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Figure 3. The efficient frontiers for different $ \beta $

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Figure 4. The efficient frontiers for different $ \lambda $

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Figure 5. The efficient frontiers for different $ \lambda_1 $

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Figure 6. The efficient frontiers for different $ \lambda_2 $

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Figure 7. The efficient frontiers for different $ \lambda_0 $

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Figure 8. The efficient frontiers for different $ \sigma $

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