Optimal investment and reinsurance of insurers with lognormal stochastic factor model

Hiroaki Hata; Li-Hsien Sun

doi:10.3934/mcrf.2021033

Article Contents

2022, Volume 12, Issue 2: 531-566. Doi: 10.3934/mcrf.2021033

This issue Previous Article Existence and cost of boundary controls for a degenerate/singular parabolic equation Next Article

Optimal investment and reinsurance of insurers with lognormal stochastic factor model

Hiroaki Hata^1, and
Li-Hsien Sun^2, ,

1.
Hitotsubashi University, Naka, Kunitachi, Tokyo 186-8601, Japan
2.
National Central University, Taoyuan City 32001, Taiwan

^* Corresponding author: Li-Hsien Sun
^* Corresponding author: Li-Hsien Sun

Received: November 2020

Revised: April 2021

Early access: June 2021

Published: June 2022

Li-Hsien Sun's research is supported by Most grant 108-2118-M-008-002-MY2.

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Abstract

We propose the stochastic factor model of optimal investment and reinsurance of insurers where the wealth processes are described by a bank account and a risk asset for investment and a Cramér-Lundberg process for reinsurance. The optimization is obtained through maximizing the exponential utility. Owing to the claims driven by a Poisson process, the proposed optimization problem is naturally treated as a jump-diffusion control problem. Applying the dynamic programming, we have the Hamilton-Jacobi-Bellman (HJB) equations and the corresponding explicit solution for the corresponding HJB. Hence, the optimal values and optimal strategies can be obtained. Finally, in numerical analysis, we illustrate the performance of the proposed optimization according to the results of the corresponding value function. In addition, compared to the wealth process without investment, the efficiency of the proposed optimization is discussed in terms of ruin probabilities.

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Mathematics Subject Classification: Primary: 93E20, 49L20, 90C40, 60H30, 91G80.

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Figure 1. The value function $ \widehat V(0,x,y) $ with $ a = b = 1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ \theta = 2 $, $ c = 1 $, $ k = 12 $, the varied parameter $ \alpha = 0.2 $

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Figure 2. The value function $ \widehat V(0,x,y) $ with $ a = b = 1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ \theta = 2 $, $ c = 1 $, $ k = 12 $, and the varied parameter $ \alpha = 0.5 $

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Figure 3. The ruin probability with $ a = b = 0.1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ \theta = 1 $, $ c = 1 $, $ k = 15 $, $ x = 10 $, and the varied $ \alpha $

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Figure 4. The ruin probability with $ a = b = 0.1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ \theta = 1 $, $ c = 1 $, $ k = 15 $, $ x = 10 $, and the relative varied small $ \alpha $

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Figure 5. The ruin probability with $ \alpha = 0.2 $ $ a = b = 0.1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ \theta = 1 $, $ c = 1 $, $ x = 10 $, and the varied $ k $

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Figure 6. The ruin probability with $ \alpha = 0.2 $ $ a = b = 0.1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \theta = 1 $, $ c = 1 $, $ k = 15 $, $ x = 10 $, and the varied $ \lambda $

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Figure 7. The ruin probability with $ \alpha = 0.2 $ $ a = b = 0.1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ c = 1 $, $ k = 15 $, $ x = 10 $, and the varied $ \theta $

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Figure 8. The ruin probability with $ \alpha = 0.2 $ $ a = b = 0.1 $, $ \beta_0 = \beta_1 = 1 $, $ r = 0.05 $, $ \rho = 0.2 $, $ \mu_0 = \mu_1 = 0.5 $, $ \lambda = 10 $, $ \theta = 1 $, $ c = 1 $, $ k = 15 $, and the varied surplus $ x $

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