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doi: 10.3934/jimo.2021145
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## Optimal per-loss reinsurance and investment to minimize the probability of drawdown

 1 Department of Statistics and Actuarial Science, University of Waterloo, Ontario, N2L 3G1, Canada 2 School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Jiangsu, 210023 China 3 School of Management Science and Engineering, Nanjing University of Information Science and Technology, Jiangsu, 210044, China 4 School of Finance, Nanjing University of Finance and Economics, Jiangsu, 210023, China

*Corresponding author: Zhibin Liang

Received  January 2021 Revised  May 2021 Early access September 2021

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant No.12071224)

In this paper, we study an optimal reinsurance-investment problem in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. We assume that the insurer can purchase per-loss reinsurance for each line of business and invest its surplus in a financial market consisting of a risk-free asset and a risky asset. Under the criterion of minimizing the probability of drawdown, the closed-form expressions for the optimal reinsurance-investment strategy and the corresponding value function are obtained. We show that the optimal reinsurance strategy is in the form of pure excess-of-loss reinsurance strategy under the expected value principle, and under the variance premium principle, the optimal reinsurance strategy is in the form of pure quota-share reinsurance. Furthermore, we extend our model to the case where the insurance company involves $n$ $(n\geq3)$ dependent classes of insurance business and the optimal results are derived explicitly as well.

Citation: Xia Han, Zhibin Liang, Yu Yuan, Caibin Zhang. Optimal per-loss reinsurance and investment to minimize the probability of drawdown. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021145
##### References:
 [1] B. Angoshtari, E. Bayraktar and V. R. Young, Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.  doi: 10.1080/17442508.20H.155590. [2] B. Angoshtari, E. Bayraktar and V. R. Young, Minimizing the probability of lifetime drawdown under constant consumption, Insurance Math. Econom., 69 (2016), 210-223.  doi: 10.1016/j.insmatheco.2016.05.007. [3] L. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance Math. Econom., 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008. [4] M. Brachetta and C. Ceci, Optimal proportional reinsurance and investment for stochastic factor models, Insurance Math. Econo., 87 (2019), 15-33.  doi: 10.1016/j.insmatheco.2019.03.006. [5] X. Chen, D. Landriault, B. Li and D. Li, On minimizing drawdown risks of lifetime investments, Insurance Math. Econom., 65 (2015), 46-54.  doi: 10.1016/j.insmatheco.2015.08.007. [6] A. Gu, X. Guo, Z. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003. [7] S. J. Grossman and Z. Zhou, Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3 (1993), 241-276. [8] X. Han, Z. Liang and V. R. Young, Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle, Scand. Actuar. J., 2020 (2020), 873-903.  doi: 10.1080/03461238.2020.1788136. [9] X. Han, Z. Liang and K. C. Yuen, Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure, Scand. Actuar. J., 2018 (2018), 863-889.  doi: 10.1080/03461238.2018.1469098. [10] X. Han, Z. Liang and C. Zhang, Optimal proportional reinsurance with common shock dependence to minimize the probability of drawdown, Annals of Actuarial Science, 13 (2019), 268-294. [11] C. Hipp and M. Taksar, Optimal non-proportional reinsurance control, Insurance Math. Econom., 47 (2010), 246-254.  doi: 10.1016/j.insmatheco.2010.04.001. [12] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance Math. Econom., 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004. [13] C. Jaksa and K. Ioannis, On portfolio optimization under drawdown constraints, IMA Lecture Notes in Mathematical Applications, 65 (1995), 77-88. [14] B. Karl, An attempt to determine the optimum amount of stop loss reinsurance, Transaction of the $16$th International Congress of Actuaries, (1960), 597–610. [15] D. Li, X. Rong and H. Zhao, Equilibrium excess-of-loss reinsurance-investment strategy for a mean-variance insurer under stochastic volatility model, Comm. Statist. Theory Methods, 46 (2017), 9459-9475.  doi: 10.1080/03610926.20I.112071. [16] D. Li, Y. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679. [17] X. Liang and V. R. Young, Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance Math. Econom., 82 (2018), 181-190.  doi: 10.1016/j.insmatheco.2018.07.005. [18] X. Liang and V. R. Young, Minimizing the discounted probability of exponential Parisian ruin via reinsurance, SIAM J. Control Optim., 58 (2020), 937-964.  doi: 10.1137/19M1282714. [19] Z. Liang, J. Bi, K. C. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Math. Methods Oper. Res., 84 (2016), 155-181.  doi: 10.1007/s00186-016-0538-0. [20] Z. Liang and J. Guo, newblock Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, J. Appl. Math. Comput., 36 (2011), 11–25., doi: 10.1007/s12190-010-0385-8. [21] S. Luo and M. Taksar, On absolute ruin minimization under a diffusion approximation model, Insurance Math. Econom., 48 (2011), 123-133. [22] V. C. Pestien and W. D. Sudderth, Continuous-time red and black: How to control a diffusion to a goal, Math. Oper. Res., 8 (1985), 599-611. [23] S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128. [24] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. [25] Y. Yuan, Z. Liang and X. Han, Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs, Journal of Industrial and Management Optimization, (2021). doi: 10.3934/jimo.2021003. [26] 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. [27] X. Zhang, M. Zhou and J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Appl. Stoch. Models Bus. Ind., 23 (2007), 63-71.

show all references

##### References:
 [1] B. Angoshtari, E. Bayraktar and V. R. Young, Optimal investment to minimize the probability of drawdown, Stochastics, 88 (2016), 946-958.  doi: 10.1080/17442508.20H.155590. [2] B. Angoshtari, E. Bayraktar and V. R. Young, Minimizing the probability of lifetime drawdown under constant consumption, Insurance Math. Econom., 69 (2016), 210-223.  doi: 10.1016/j.insmatheco.2016.05.007. [3] L. Bai, J. Cai and M. Zhou, Optimal reinsurance policies for an insurer with a bivariate reserve risk process in a dynamic setting, Insurance Math. Econom., 53 (2013), 664-670.  doi: 10.1016/j.insmatheco.2013.09.008. [4] M. Brachetta and C. Ceci, Optimal proportional reinsurance and investment for stochastic factor models, Insurance Math. Econo., 87 (2019), 15-33.  doi: 10.1016/j.insmatheco.2019.03.006. [5] X. Chen, D. Landriault, B. Li and D. Li, On minimizing drawdown risks of lifetime investments, Insurance Math. Econom., 65 (2015), 46-54.  doi: 10.1016/j.insmatheco.2015.08.007. [6] A. Gu, X. Guo, Z. Li and Y. Zeng, Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model, Insurance Math. Econom., 51 (2012), 674-684.  doi: 10.1016/j.insmatheco.2012.09.003. [7] S. J. Grossman and Z. Zhou, Optimal investment strategies for controlling drawdowns, Mathematical Finance, 3 (1993), 241-276. [8] X. Han, Z. Liang and V. R. Young, Optimal reinsurance to minimize the probability of drawdown under the mean-variance premium principle, Scand. Actuar. J., 2020 (2020), 873-903.  doi: 10.1080/03461238.2020.1788136. [9] X. Han, Z. Liang and K. C. Yuen, Optimal proportional reinsurance to minimize the probability of drawdown under thinning-dependence structure, Scand. Actuar. J., 2018 (2018), 863-889.  doi: 10.1080/03461238.2018.1469098. [10] X. Han, Z. Liang and C. Zhang, Optimal proportional reinsurance with common shock dependence to minimize the probability of drawdown, Annals of Actuarial Science, 13 (2019), 268-294. [11] C. Hipp and M. Taksar, Optimal non-proportional reinsurance control, Insurance Math. Econom., 47 (2010), 246-254.  doi: 10.1016/j.insmatheco.2010.04.001. [12] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance Math. Econom., 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004. [13] C. Jaksa and K. Ioannis, On portfolio optimization under drawdown constraints, IMA Lecture Notes in Mathematical Applications, 65 (1995), 77-88. [14] B. Karl, An attempt to determine the optimum amount of stop loss reinsurance, Transaction of the $16$th International Congress of Actuaries, (1960), 597–610. [15] D. Li, X. Rong and H. Zhao, Equilibrium excess-of-loss reinsurance-investment strategy for a mean-variance insurer under stochastic volatility model, Comm. Statist. Theory Methods, 46 (2017), 9459-9475.  doi: 10.1080/03610926.20I.112071. [16] D. Li, Y. Zeng and H. Yang, Robust optimal excess-of-loss reinsurance and investment strategy for an insurer in a model with jumps, Scand. Actuar. J., 2018 (2018), 145-171.  doi: 10.1080/03461238.2017.1309679. [17] X. Liang and V. R. Young, Minimizing the probability of ruin: Optimal per-loss reinsurance, Insurance Math. Econom., 82 (2018), 181-190.  doi: 10.1016/j.insmatheco.2018.07.005. [18] X. Liang and V. R. Young, Minimizing the discounted probability of exponential Parisian ruin via reinsurance, SIAM J. Control Optim., 58 (2020), 937-964.  doi: 10.1137/19M1282714. [19] Z. Liang, J. Bi, K. C. Yuen and C. Zhang, Optimal mean-variance reinsurance and investment in a jump-diffusion financial market with common shock dependence, Math. Methods Oper. Res., 84 (2016), 155-181.  doi: 10.1007/s00186-016-0538-0. [20] Z. Liang and J. Guo, newblock Optimal combining quota-share and excess of loss reinsurance to maximize the expected utility, J. Appl. Math. Comput., 36 (2011), 11–25., doi: 10.1007/s12190-010-0385-8. [21] S. Luo and M. Taksar, On absolute ruin minimization under a diffusion approximation model, Insurance Math. Econom., 48 (2011), 123-133. [22] V. C. Pestien and W. D. Sudderth, Continuous-time red and black: How to control a diffusion to a goal, Math. Oper. Res., 8 (1985), 599-611. [23] S. D. Promislow and V. R. Young, Minimizing the probability of ruin when claims follow Brownian motion with drift, N. Am. Actuar. J., 9 (2005), 109-128. [24] H. Schmidli, On minimizing the ruin probability by investment and reinsurance, Ann. Appl. Probab., 12 (2002), 890-907. [25] Y. Yuan, Z. Liang and X. Han, Optimal investment and reinsurance to minimize the probability of drawdown with borrowing costs, Journal of Industrial and Management Optimization, (2021). doi: 10.3934/jimo.2021003. [26] 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. [27] X. Zhang, M. Zhou and J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Appl. Stoch. Models Bus. Ind., 23 (2007), 63-71.
The function $G(7,d)$
The effect of $u$ on optimal strategy
The effect of η1 on optimal strategy (η2 = 0.25)
The effect of $\eta_2$ on optimal strategy ($\eta_1 = 0.4$)
The effect of $\lambda$ on optimal strategy
The effect of $\lambda$ on correlation coefficient
Comparison between the optimal strategies
Comparison between the value functions
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