# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021033
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## Optimal investment and reinsurance of insurers with lognormal stochastic factor model

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

* Corresponding author: Li-Hsien Sun

Received  November 2020 Revised  April 2021 Early access June 2021

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

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.

Citation: Hiroaki Hata, Li-Hsien Sun. Optimal investment and reinsurance of insurers with lognormal stochastic factor model. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021033
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##### References:
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 $\alpha = 0.2$
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 $\alpha = 0.5$
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 $\alpha$
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 $\alpha$
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 $k$
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 $\lambda$
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 $\theta$
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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