Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs

Yu Yuan; Zhibin Liang; Xia Han

doi:10.3934/jimo.2021003

Article Contents

2022, Volume 18, Issue 2: 933-967. Doi: 10.3934/jimo.2021003

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Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs

School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu 210023, China

^* Corresponding author: Zhibin Liang
^* Corresponding author: Zhibin Liang

Received: January 2020

Revised: September 2020

Early access: December 2020

Published: March 2022

This research was supported by the National Natural Science Foundation of China (Grant No.12071224).

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Abstract

We study the optimal investment and reinsurance problem in a risk model with two dependent classes of insurance businesses, where the two claim number processes are correlated through a common shock component and the borrowing rate is higher than the lending rate. The objective is to minimize the probability of drawdown, namely, the probability that the value of the wealth process reaches some fixed proportion of its maximum value to date. By the method of stochastic control theory and the corresponding Hamilton-Jacobi-Bellman equation, we investigate the optimization problem in two different cases and divide the whole region into four subregions. The explicit expressions for the optimal investment/reinsurance strategies and the minimum probability of drawdown are derived. We find that when wealth is at a relatively low level (below the borrowing level), it is optimal to borrow money to invest in the risky asset; when wealth is at a relatively high level (above the saving level), it is optimal to save more money; while between them, the insurer is willing to invest all the wealth in the risky asset. In the end, some comparisons are presented to show the impact of higher borrowing rate and risky investment on the optimal results.

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Mathematics Subject Classification: Primary: 93E20; Secondary: 62P05, 91G05, 91G10.

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Figure 1. The influence of higher borrowing rate on the optimal investment strategies

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Figure 2. The influence of higher borrowing rate on the optimal reinsurance strategies

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Figure 3. The influence of risky investment on the optimal reinsurance strategies

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