Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints

Liming Zhang; Rongming Wang; Jiaqin Wei

doi:10.3934/jimo.2021140

Article Contents

2022, Volume 18, Issue 6: 3897-3927. Doi: 10.3934/jimo.2021140

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Open-loop equilibrium mean-variance reinsurance, new business and investment strategies with constraints

1.
School of Statistics, East China Normal University, Shanghai 200062, China
2.
Key Laboratory of Advanced Theory and, Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai 200241, China

^* Corresponding author: jqwei@stat.ecnu.edu.cn
^* Corresponding author: jqwei@stat.ecnu.edu.cn

Received: February 28, 2021

Revised: April 30, 2021

Early access: August 2021

Published: September 2022

This work was supported by the Central Universities (2020ECNU-HWFW003), the 111 Project (B14019); the National Natural Science Foundation of China (12071147, 12071146, 11571113, 11601157, 11601320); the China Scholarship Council (201906140044); Graduate Research Innovation Program of School of Statistics, East China Normal University

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Abstract

In this paper, we study a general mean-variance reinsurance, new business and investment problem, where the claim processes of original and new businesses are modeled by two different risk processes and the safety loadings of reinsurance and new business are different. The retention level of the insurer is constrained in $ [0,1] $ and the controls of new business and risky investment are required to be non-negative. This model relaxes the limitations of those in existing research. By using the projection onto the convex set controls valued in, we obtain an open-loop equilibrium reinsurance-new business-investment strategy explicitly. We also show that the obtained equilibrium strategy is the optimal one among all deterministic strategies in the sense that it yields the smallest mean-variance cost. In the case where original and new businesses are the same, the equilibrium strategy is given in closed-form and its sensitivities to safety loadings are shown by numerical examples. At last, by comparing with the case where acquiring new business is prohibited, we show that allowing writing new policies indeed improves the performance of the insurer's risk management.

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Mathematics Subject Classification: Primary: 90B50, 93E20; Secondary: 91G80.

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Figure 1. Equilibrium strategies, $a_{1} = 0.2, \ b_{1} = 0.4, \ a_{2} = 0.3, \ b_{21} = 0, \ b_{22} = 0.5, \ r = 0.05, \ \mu = 0.2, \ \sigma_{1} = \sigma_{2} = 0, \ \sigma_{3} = 0.3, \ T = 10, \ \lambda = 1 $

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Figure 2. Equilibrium retention level and new business strategy, $a_{1} = a_{2} = 0.2, \ b_{1} = b_{21} = 0.4, \ b_{22} = 0, \ r = 0.05, \ \mu = 0.2, \ \sigma_{2} = 0, \ \sigma_{3} = 0.3, \ T = 10, \ \lambda = 1 $

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Figure 3. Equilibrium investment strategy, $a_{1} = a_{2} = 0.2, \ b_{1} = b_{21} = 0.4, \ b_{22} = 0, \ r = 0.05, \ \mu = 0.2, \ \sigma_{2} = 0, \ \sigma_{3} = 0.3, \ T = 10, \ \lambda = 1$

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Table 1. Relations among $ x_{i} $, $ i = 1,2,3,4 $

Cases	Relations
$ \sigma_{1}>\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}} $	$ 0 <x_{1}<x_{3}<x_{2}<x_{4} $
$ \sigma_{1}=\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}} $	$ 0<x_{1}<x_{3}=x_{2}=x_{4} $
$ \sigma_{1}\in\left(0,\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}\right) $	$ 0<x_{1}<x_{4}<x_{2}<x_{3} $
$ \sigma_=0$	$ 0=x_{1}<x_{2}=x_{4} $
$ \sigma_{1}\in\left(-\frac{\sigma_{3}^{2}b_{1}P_{0}}{\lambda\left(\mu-r\right)},0\right) $	$ x_{3}<x_{1}<0<x_{2}<x_{4} $
$ \sigma_{1}\leq-\frac{\sigma_{3}^{2}b_{1}P_{0}}{\lambda\left(\mu-r\right)} $	$ x_{3}<x_{1}<x_{2}\leq0<x_{4} $

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Table 2. Comparisons of $ \widehat{\boldsymbol{u}}_{1}$ and $\boldsymbol{u}_{1}^{*}$

Cases		$p^{\ast}- \widehat{p}$	$\pi^{\ast}- \widehat{\pi}$
	$\eta\leq x_{1}$	0	$\frac{\lambda a_{1}\sigma_{1}\left(x_{1}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\geq0$
$\sigma_{1}\geq\frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}$	$\eta\in\left(x_{1}, x_{2}\right)$	0	0
	$\eta\in\left[x_{3}, x_{2}\right)$	$\frac{\lambda a_{1}\sigma_{1}^{2}\left(\eta-x_{3}\right)}{\sigma_{3}^{2}b_{1}^{2}P_{0}}\geq0$	0
	$\eta\in\left[x_{2}, x_{4}\right)$	$\frac{\lambda a_{1}\left(x_{4}-\eta\right)}{b_{1}^{2} P_{0}}>0$	0
	$\eta\geq x_{4}$	0	0
	$\eta\leq x_{1}$	0	$\frac{\lambda a_{1}\sigma_{1}\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)\left(x_{1}-\eta\right)}{\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)\sigma_{3}^{2}b_{1}P_{0}}\geq0$
$\sigma_{1}\in\left(0, \frac{\lambda\left(\mu-r\right)}{b_{1}P_{0}}\right)$	$\eta\in\left(x_{1}, x_{2}\right)$	0	0
	$\eta\in\left[x_{2}, x_{3}\right)$	0	$\frac{\lambda a_{1}\sigma_{1}\left(x_{2}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\leq0$
	$\eta\geq x_{3}$	0	$\frac{\sigma_{1}b_{1}P_{0}-\lambda\left(\mu-r\right)}{\left(\sigma_{1}^{2}+\sigma_{3}^{2}\right)P_{0}} <0$
$\sigma_{1}\leq0$	$\eta<x_{2}$	0	0
	$\eta\geq x_{2}$	0	$\frac{\lambda a_{1}\sigma_{1}\left(x_{2}-\eta\right)}{\sigma_{3}^{2}b_{1}P_{0}}\leq0$

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