A Stackelberg reinsurance-investment game under Heston's stochastic volatility model

Guoyong Zhou; Zhijian Qiu; Sheng Li

doi:10.3934/jimo.2022133

Article Contents

2023, Volume 19, Issue 6: 4350-4380. Doi: 10.3934/jimo.2022133

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A Stackelberg reinsurance-investment game under Heston's stochastic volatility model

1.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, China
2.
School of Statistics, Chengdu University of Information Technology, Chengdu, 610103, China

^*Corresponding author: Sheng Li
^*Corresponding author: Sheng Li

Received: October 2021

Revised: June 2022

Early access: August 2022

Published: June 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

This paper studies the optimal reinsurance-investment problem under Heston's stochastic volatility (SV) in the framework of Stackelberg stochastic differential game, in which an insurer and a reinsurer are the two players. The aim of the insurer as the follower in the game is to find the optimal reinsurance strategy and investment strategy such that its mean-variance cost functional is maximized, while the aim of the reinsurer as the leader is to maximize its mean-variance cost functional through finding its optimal premium pricing strategy and investment strategy. To overcome the time-inconsistency problem in the game, we derive the optimization problems of the two players as embedded games and obtain the corresponding extended Hamilton–Jacobi–Bellman (HJB) equations. Then we present the verification theorem and get the equilibrium reinsurance-investment strategies and the corresponding value functions of both the insurer and the reinsurer. Finally, we provide some numerical examples to draw the findings.

Keywords:

Stackelberg stochastic differential game,
reinsurance and investment strategy,
mean-variance criterion,
relative performance.

Mathematics Subject Classification: Primary: 91A23, 93E20; Secondary: 97M30.

Citation:

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Figure 1. The effects of $ k $ and $ m_1 $ on $ \pi_2^*(0) $

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Figure 2. The effects of $ k $ and $ m_2 $ on $ \pi_2^*(0) $

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Figure 3. The effects of $ k $ and $ m_1 $ on $ \pi_1^*(0) $

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Figure 4. The effects of $ k $ and $ m_2 $ on $ \pi_1^*(0) $

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Figure 5. The effects of time horizon $ T $ on $ \pi_i^*(0) $, for $ i\in{1, 2} $

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Figure 6. The effects of decision time $ t $ on $ \pi_i^*(t) $, for $ i\in{1, 2} $

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Figure 7. Effects of time $ t $ on $ p^*(t) $and $ q^*(t) $

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Table 1. The optimal premium strategy and reinsurance strategy in the insurance market under different cases

Cases	$ p^*(t) $	$ q^*(t) $
(1)$ H^\theta\geq 1 $	$ \forall p\in [c, \bar c] $	1
(2) $ H^{\bar\theta}\leq H $	$ \bar c $	$ \frac{H^{\bar\theta}+k}{1+k} $
(3)$ H\leq H^{\theta}<1 $	$ c $	$ \frac{H^\theta+k}{1+k} $
(4)$ H^\theta<H<H^{\bar\theta} $	$ \lambda\mu+m_1\lambda\sigma_1^2D(t)H $	$ \frac{(1+2k)m_1+m_2}{2(1+k)m_1+m_2} $

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Table 2. Model parameters

Financial market
	$ t $	$ T $	$ r $	$ \alpha $	$ \beta $	$ \delta $	$ \sigma_2 $	$ \rho $	$ s_0 $	$ l_0 $
	0	10	0.05	1.5	2	0.03	1.5	0.3	1	1
Insurance market
	$ \mu $	$ \lambda $	$ \theta $	$ \bar\theta $	$ \sigma_1 $	$ m_1 $	$ m_2 $	$ k $
	2	3	0.1	0.2	1	0.6	0.4	0.4

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Table 3. Numerical results corresponding to Figure 7

$ t $	0	1	2	3	4	5	6	7	8	9	10
$ H $	0.5962	0.5962	0.5962	0.5962	0.5962	0.5962	0.5962	0.5962	0.5962	0.5962	0.5962
$ H^{\theta} $	0.2125	0.2234	0.2349	0.2469	0.2596	0.2729	0.2869	0.3016	0.3171	0.3333	0.3504
$ H^{\bar\theta} $	0.4250	0.4469	0.4698	0.4939	0.5192	0.5458	0.5738	0.6032	0.6342	0.6667	0.7008
$ p^*(t) $	7.2000	7.2000	7.2000	7.2000	7.2000	7.2000	7.2000	7.1859	7.1281	7.0731	7.0207
$ q^*(t) $	0.5893	0.6049	0.6213	0.6385	0.6566	0.6756	0.6956	0.7115	0.7115	0.7115	0.7115

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