# American Institute of Mathematical Sciences

• Previous Article
On the asymptotic stability of the Korteweg-de Vries equation with time-delayed internal feedback
• MCRF Home
• This Issue
• Next Article
Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions
doi: 10.3934/mcrf.2021033
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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
##### References:

show all references

##### 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 $x$
 [1] Yan Zhang, Yonghong Wu, Benchawan Wiwatanapataphee, Francisca Angkola. Asset liability management for an ordinary insurance system with proportional reinsurance in a CIR stochastic interest rate and Heston stochastic volatility framework. Journal of Industrial & Management Optimization, 2020, 16 (1) : 71-101. doi: 10.3934/jimo.2018141 [2] Adel Chala, Dahbia Hafayed. On stochastic maximum principle for risk-sensitive of fully coupled forward-backward stochastic control of mean-field type with application. Evolution Equations & Control Theory, 2020, 9 (3) : 817-843. doi: 10.3934/eect.2020035 [3] Mrinal K. Ghosh, Somnath Pradhan. A nonzero-sum risk-sensitive stochastic differential game in the orthant. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021025 [4] Lihua Bian, Zhongfei Li, Haixiang Yao. Time-consistent strategy for a multi-period mean-variance asset-liability management problem with stochastic interest rate. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1383-1410. doi: 10.3934/jimo.2020026 [5] Chandan Pal, Somnath Pradhan. Zero-sum games for pure jump processes with risk-sensitive discounted cost criteria. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021020 [6] Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial & Management Optimization, 2020, 16 (2) : 813-834. doi: 10.3934/jimo.2018180 [7] Haixiang Yao, Zhongfei Li, Yongzeng Lai. Dynamic mean-variance asset allocation with stochastic interest rate and inflation rate. Journal of Industrial & Management Optimization, 2016, 12 (1) : 187-209. doi: 10.3934/jimo.2016.12.187 [8] Jiongmin Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control & Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271 [9] Lin Xu, Rongming Wang. Upper bounds for ruin probabilities in an autoregressive risk model with a Markov chain interest rate. Journal of Industrial & Management Optimization, 2006, 2 (2) : 165-175. doi: 10.3934/jimo.2006.2.165 [10] María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004 [11] Haiyang Wang, Zhen Wu. Time-inconsistent optimal control problem with random coefficients and stochastic equilibrium HJB equation. Mathematical Control & Related Fields, 2015, 5 (3) : 651-678. doi: 10.3934/mcrf.2015.5.651 [12] Andrew P. Sage. Risk in system of systems engineering and management. Journal of Industrial & Management Optimization, 2008, 4 (3) : 477-487. doi: 10.3934/jimo.2008.4.477 [13] Qigang Yuan, Yutong Sun, Jingli Ren. How interest rate influences a business cycle model. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3231-3251. doi: 10.3934/dcdss.2020190 [14] Zhongbao Zhou, Ximei Zeng, Helu Xiao, Tiantian Ren, Wenbin Liu. Multiperiod portfolio optimization for asset-liability management with quadratic transaction costs. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1493-1515. doi: 10.3934/jimo.2018106 [15] Yu Yuan, Hui Mi. Robust optimal asset-liability management with penalization on ambiguity. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021121 [16] Srdjan Stojanovic. Interest rates risk-premium and shape of the yield curve. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1603-1615. doi: 10.3934/dcdsb.2016013 [17] Lili Ding, Xinmin Liu, Yinfeng Xu. Competitive risk management for online Bahncard problem. Journal of Industrial & Management Optimization, 2010, 6 (1) : 1-14. doi: 10.3934/jimo.2010.6.1 [18] Linlin Tian, Xiaoyi Zhang, Yizhou Bai. Optimal dividend of compound poisson process under a stochastic interest rate. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2141-2157. doi: 10.3934/jimo.2019047 [19] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [20] Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011

2020 Impact Factor: 1.284