# American Institute of Mathematical Sciences

June  2022, 12(2): 531-566. doi: 10.3934/mcrf.2021033

## 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 Published  June 2022 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 and Related Fields, 2022, 12 (2) : 531-566. doi: 10.3934/mcrf.2021033
##### References:
 [1] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503. [3] M. Badaoui and B. Fernández, An optimal investment strategy with maximal risk aversion and its ruin probability in the presence of stochastic volatility on investments, Insurance Math. Econom., 53 (2013), 1-13.  doi: 10.1016/j.insmatheco.2013.04.002. [4] M. Badaoui, B. Fernández and A. Swishchuk, An optimal investment strategy for insurers in incomplete markets, Risks, 6 (2018), 31pp. doi: 10.3390/risks6020031. [5] L. Bo and S. Wang, Optimal investment and risk control for an insurer with stochastic factor, Oper. Res. Lett., 45 (2017), 259-265.  doi: 10.1016/j.orl.2017.04.002. [6] T. S. Ferguson, Betting systems which minimize the probability of ruin, J. Soc. Indust. Appl. Math., 13 (1965), 795-818.  doi: 10.1137/0113051. [7] B. Fernández, D. Hernández, A. Meda and P. Saavedra, An optimal investment strategy with maximal risk aversion and its ruin probability, Math. Methods Oper. Res., 68 (2008), 159-179.  doi: 10.1007/s00186-007-0191-8. [8] W. H. Fleming and M. Soner, Control Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. [9] G. Guan and Z. Liang, Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks, Insurance Math. Econom., 55 (2014), 105-115.  doi: 10.1016/j.insmatheco.2014.01.007. [10] M. Guerra and M. L. Centeno, Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria, Insurance Math. Econom., 42 (2008), 529-539.  doi: 10.1016/j.insmatheco.2007.02.008. [11] H. Hata and J. Sekine, Solving long term optimal investment problems with Cox-Ingersoll-Ross interest rates, Adv. Math. Econ., 8 (2006), 231-255.  doi: 10.1007/4-431-30899-7_9. [12] H. Hata and K. Yasuda, Expected exponential utility maximization of insurers with a linear Gaussian stochastic factor model, Scand. Actuar. J., 5 (2018), 357-378.  doi: 10.1080/03461238.2017.1350876. [13] Z. Liang and E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insurance Math. Econom., 55 (2014), 156-166.  doi: 10.1016/j.insmatheco.2014.01.011. [14] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, J. Comput. Appl. Math., 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032. [15] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk, Insurance Math. Econom., 64 (2015), 28-44.  doi: 10.1016/j.insmatheco.2015.05.003. [16] T. Li and Z. Li, Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion, Insurance Math. Econom., 53 (2013), 86-97.  doi: 10.1016/j.insmatheco.2013.03.008. [17] Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002. [18] Z. Liang, K. C. Yuen and J. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance Math Econom., 49 (2011), 207-215.  doi: 10.1016/j.insmatheco.2011.04.005. [19] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173. [20] Y. Shen and Y. Zeng, Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance Math. Econom., 62 (2015), 118-137.  doi: 10.1016/j.insmatheco.2015.03.009. [21] D. L. Sheng, X. Rong and H. Zhao, Optimal control of investment-reinsurance problem for an insurer with jump-diffusion risk process: Independence of brownian motions, Abstr. Appl. Anal., 2014 (2014), Art. ID 194962, 19 pp. doi: 10.1155/2014/194962. [22] N. Wang, Optimal investment for an insurer with exponential utility preference, Insurance Math. Econom., 40 (2007), 77-84.  doi: 10.1016/j.insmatheco.2006.02.008. [23] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001. [24] Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance Math. Econom., 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003. [25] L. Xu, L. Zhang and D. Yao, Optimal investment and reinsurance for an insurer under markov-modulated financial market, Insurance Math. Econom., 74 (2017), 7-19.  doi: 10.1016/j.insmatheco.2017.02.005. [26] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [27] X. Zhang, H. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance Math. Econom., 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001. [28] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004. [29] Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001. [30] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006. [31] Q. Zhou, Optimal investment for an insurer in the Lévy market: The martingale approach, Statistics and Probability Letters, 79 (2009), 1602-1607.  doi: 10.1016/j.spl.2009.03.027. [32] M. Yor, On some exponential functionals of Brownian motion, Adv. Appl. Prob., 24 (1992), 509-531.  doi: 10.2307/1427477.

show all references

##### References:
 [1] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance Math. Econom., 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [2] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511526503. [3] M. Badaoui and B. Fernández, An optimal investment strategy with maximal risk aversion and its ruin probability in the presence of stochastic volatility on investments, Insurance Math. Econom., 53 (2013), 1-13.  doi: 10.1016/j.insmatheco.2013.04.002. [4] M. Badaoui, B. Fernández and A. Swishchuk, An optimal investment strategy for insurers in incomplete markets, Risks, 6 (2018), 31pp. doi: 10.3390/risks6020031. [5] L. Bo and S. Wang, Optimal investment and risk control for an insurer with stochastic factor, Oper. Res. Lett., 45 (2017), 259-265.  doi: 10.1016/j.orl.2017.04.002. [6] T. S. Ferguson, Betting systems which minimize the probability of ruin, J. Soc. Indust. Appl. Math., 13 (1965), 795-818.  doi: 10.1137/0113051. [7] B. Fernández, D. Hernández, A. Meda and P. Saavedra, An optimal investment strategy with maximal risk aversion and its ruin probability, Math. Methods Oper. Res., 68 (2008), 159-179.  doi: 10.1007/s00186-007-0191-8. [8] W. H. Fleming and M. Soner, Control Markov Processes and Viscosity Solutions, Second edition. Stochastic Modelling and Applied Probability, 25. Springer, New York, 2006. [9] G. Guan and Z. Liang, Optimal reinsurance and investment strategies for insurer under interest rate and inflation risks, Insurance Math. Econom., 55 (2014), 105-115.  doi: 10.1016/j.insmatheco.2014.01.007. [10] M. Guerra and M. L. Centeno, Optimal reinsurance policy: The adjustment coefficient and the expected utility criteria, Insurance Math. Econom., 42 (2008), 529-539.  doi: 10.1016/j.insmatheco.2007.02.008. [11] H. Hata and J. Sekine, Solving long term optimal investment problems with Cox-Ingersoll-Ross interest rates, Adv. Math. Econ., 8 (2006), 231-255.  doi: 10.1007/4-431-30899-7_9. [12] H. Hata and K. Yasuda, Expected exponential utility maximization of insurers with a linear Gaussian stochastic factor model, Scand. Actuar. J., 5 (2018), 357-378.  doi: 10.1080/03461238.2017.1350876. [13] Z. Liang and E. Bayraktar, Optimal reinsurance and investment with unobservable claim size and intensity, Insurance Math. Econom., 55 (2014), 156-166.  doi: 10.1016/j.insmatheco.2014.01.011. [14] Y. Huang, X. Yang and J. Zhou, Optimal investment and proportional reinsurance for a jump-diffusion risk model with constrained control variables, J. Comput. Appl. Math., 296 (2016), 443-461.  doi: 10.1016/j.cam.2015.09.032. [15] D. Li, X. Rong and H. Zhao, Time-consistent reinsurance-investment strategy for a mean-variance insurer under stochastic interest rate model and inflation risk, Insurance Math. Econom., 64 (2015), 28-44.  doi: 10.1016/j.insmatheco.2015.05.003. [16] T. Li and Z. Li, Optimal time-consistent investment and reinsurance strategies for mean-variance insurers with state dependent risk aversion, Insurance Math. Econom., 53 (2013), 86-97.  doi: 10.1016/j.insmatheco.2013.03.008. [17] Z. Li, Y. Zeng and Y. Lai, Optimal time-consistent investment and reinsurance strategies for insurers under Heston's SV model, Insurance Math. Econom., 51 (2012), 191-203.  doi: 10.1016/j.insmatheco.2011.09.002. [18] Z. Liang, K. C. Yuen and J. Guo, Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process, Insurance Math Econom., 49 (2011), 207-215.  doi: 10.1016/j.insmatheco.2011.04.005. [19] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173. [20] Y. Shen and Y. Zeng, Optimal investment-reinsurance strategy for mean-variance insurers with square-root factor process, Insurance Math. Econom., 62 (2015), 118-137.  doi: 10.1016/j.insmatheco.2015.03.009. [21] D. L. Sheng, X. Rong and H. Zhao, Optimal control of investment-reinsurance problem for an insurer with jump-diffusion risk process: Independence of brownian motions, Abstr. Appl. Anal., 2014 (2014), Art. ID 194962, 19 pp. doi: 10.1155/2014/194962. [22] N. Wang, Optimal investment for an insurer with exponential utility preference, Insurance Math. Econom., 40 (2007), 77-84.  doi: 10.1016/j.insmatheco.2006.02.008. [23] Y. Wang, X. Rong and H. Zhao, Optimal investment strategies for an insurer and a reinsurer with a jump diffusion risk process under the CEV model, J. Comput. Appl. Math., 328 (2018), 414-431.  doi: 10.1016/j.cam.2017.08.001. [24] Z. Wang, J. Xia and L. Zhang, Optimal investment for an insurer: The martingale approach, Insurance Math. Econom., 40 (2007), 322-334.  doi: 10.1016/j.insmatheco.2006.05.003. [25] L. Xu, L. Zhang and D. Yao, Optimal investment and reinsurance for an insurer under markov-modulated financial market, Insurance Math. Econom., 74 (2017), 7-19.  doi: 10.1016/j.insmatheco.2017.02.005. [26] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance Math. Econom., 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [27] X. Zhang, H. Meng and Y. Zeng, Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance Math. Econom., 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001. [28] H. Zhao, X. Rong and Y. Zhao, Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model, Insurance Math. Econom., 53 (2013), 504-514.  doi: 10.1016/j.insmatheco.2013.08.004. [29] Y. Zeng and Z. Li, Optimal time-consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145-154.  doi: 10.1016/j.insmatheco.2011.01.001. [30] B. Zou and A. Cadenillas, Optimal investment and risk control policies for an insurer: Expected utility maximization, Insurance Math. Econom., 58 (2014), 57-67.  doi: 10.1016/j.insmatheco.2014.06.006. [31] Q. Zhou, Optimal investment for an insurer in the Lévy market: The martingale approach, Statistics and Probability Letters, 79 (2009), 1602-1607.  doi: 10.1016/j.spl.2009.03.027. [32] M. Yor, On some exponential functionals of Brownian motion, Adv. Appl. Prob., 24 (1992), 509-531.  doi: 10.2307/1427477.
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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