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October  2017, 13(4): 1771-1791. doi: 10.3934/jimo.2017018

## Minimizing expected time to reach a given capital level before ruin

 1 School of Sciences, Hebei University of Technology, Tianjin 300401, China 2 School of Mathematical Sciences, Nankai University, Tianjin 300071, China

* Corresponding author: Lihua Bai

Received  November 2015 Published  December 2016

Fund Project: The first author is supported by the National Natural Science Foundation of China (11571189),the project RARE -318984 (an FP7 Marie Curie IRSES) and High School National Science Foundation of Hebei Province (QN2016176), and the second author is supported by the National Natural Science Foundation of China (11471171).

In this paper, we consider the optimal investment and reinsurance problem for an insurance company where the claim process follows a Brownian motion with drift. The insurer can purchase proportional reinsurance and invest its surplus in one risky asset and one risk-free asset. The goal of the insurance company is to minimize the expected time to reach a given capital level before ruin. By using the Hamilton-Jacobi-Bellman equation approach, we obtain explicit expressions for the value function and the optimal strategy. We also provide some numerical examples to illustrate the results obtained in this paper, and analyze the sensitivity of the parameters.

Citation: Xiaoqing Liang, Lihua Bai. Minimizing expected time to reach a given capital level before ruin. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1771-1791. doi: 10.3934/jimo.2017018
##### References:
 [1] S. Asmussen, B. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075. [2] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0. [3] P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x. [4] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [5] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [6] L. Chen and H. Yang, Optimal reinsurance and investment strategy with two piece utility function Journal of Industrial and Management Optimization, 12 (2016). doi: 10.3934/jimo.2016044. [7] T. Choulli, M. Taksar and X. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.  doi: 10.1137/S0363012900382667. [8] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, 2006. [9] J. Grandell, Aspects of Risk Theory, Springer, 1991. doi: 10.1007/978-1-4613-9058-9. [10] B. Højgaard and M. Taksar, Controlling risk exposure and dividends pay-out schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066. [11] B. Højgaard and M. Taksar, Optimal risk control for a large corporation in the presence of returns on investments, Finance and Stochastics, 5 (2001), 527-547.  doi: 10.1007/PL00000042. [12] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007. [13] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004. [14] Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005. [15] Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scandinavian Actuarial Journal, 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899. [16] S. Luo, M. Wang and X. Zeng, Optimal reinsurance: Minimize the expected time to reach a goal, Scandinavian Actuarial Journal, 2016 (2015), 741-762.  doi: 10.1080/03461238.2015.1015161. [17] J. Paulsen, Optimal dividend payouts for diffusions with solvency constraints, Finance and Stochastics, 7 (2003), 457-473.  doi: 10.1007/s007800200098. [18] J. Paulsen and H. Gjessing, Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance: Mathematics and Economics, 20 (1997), 215-223.  doi: 10.1016/S0167-6687(97)00011-5. [19] V. Pestien and W. Sudderth, Continuous-time red and black: how to control a diffusion to a goal, Mathematics of Operations Research, 10 (1985), 599-611.  doi: 10.1287/moor.10.4.599. [20] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68.  doi: 10.1080/034612301750077338. [21] M. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7 (2003), 97-121.  doi: 10.1007/s007800200073. [22] N. Wang, Optimal investment for an insurer with exponential utility preference, Insurance: Mathematics and Economics, 40 (2007), 77-84.  doi: 10.1016/j.insmatheco.2006.02.008. [23] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [24] C. Yin and K. C. Yuen, Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs, Journal of Industrial and Management Optimization, 11 (2015), 1247-1262.  doi: 10.3934/jimo.2015.11.1247. [25] X. Zhang, M. Zhou and J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Applied Stochastic Models in Business and Industry, 23 (2007), 63-71.  doi: 10.1002/asmb.637. [26] M. Zhou and K. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.

show all references

##### References:
 [1] S. Asmussen, B. Højgaard and M. Taksar, Optimal risk control and dividend distribution policies: Example of excess-of-loss reinsurance for an insurance corporation, Finance and Stochastics, 4 (2000), 299-324.  doi: 10.1007/s007800050075. [2] S. Asmussen and M. Taksar, Controlled diffusion models for optimal dividend pay-out, Insurance: Mathematics and Economics, 20 (1997), 1-15.  doi: 10.1016/S0167-6687(96)00017-0. [3] P. Azcue and N. Muler, Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Mathematical Finance, 15 (2005), 261-308.  doi: 10.1111/j.0960-1627.2005.00220.x. [4] L. Bai and J. Guo, Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42 (2008), 968-975.  doi: 10.1016/j.insmatheco.2007.11.002. [5] S. Browne, Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin, Mathematics of Operations Research, 20 (1995), 937-958.  doi: 10.1287/moor.20.4.937. [6] L. Chen and H. Yang, Optimal reinsurance and investment strategy with two piece utility function Journal of Industrial and Management Optimization, 12 (2016). doi: 10.3934/jimo.2016044. [7] T. Choulli, M. Taksar and X. Zhou, A diffusion model for optimal dividend distribution for a company with constraints on risk control, SIAM Journal on Control and Optimization, 41 (2003), 1946-1979.  doi: 10.1137/S0363012900382667. [8] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer Science & Business Media, 2006. [9] J. Grandell, Aspects of Risk Theory, Springer, 1991. doi: 10.1007/978-1-4613-9058-9. [10] B. Højgaard and M. Taksar, Controlling risk exposure and dividends pay-out schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066. [11] B. Højgaard and M. Taksar, Optimal risk control for a large corporation in the presence of returns on investments, Finance and Stochastics, 5 (2001), 527-547.  doi: 10.1007/PL00000042. [12] B. Højgaard and M. Taksar, Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy, Quantitative Finance, 4 (2004), 315-327.  doi: 10.1088/1469-7688/4/3/007. [13] C. Irgens and J. Paulsen, Optimal control of risk exposure, reinsurance and investments for insurance portfolios, Insurance: Mathematics and Economics, 35 (2004), 21-51.  doi: 10.1016/j.insmatheco.2004.04.004. [14] Z. Liang and V. Young, Dividends and reinsurance under a penalty for ruin, Insurance: Mathematics and Economics, 50 (2012), 437-445.  doi: 10.1016/j.insmatheco.2012.02.005. [15] Z. Liang and K. Yuen, Optimal dynamic reinsurance with dependent risks: Variance premium principle, Scandinavian Actuarial Journal, 2016 (2016), 18-36.  doi: 10.1080/03461238.2014.892899. [16] S. Luo, M. Wang and X. Zeng, Optimal reinsurance: Minimize the expected time to reach a goal, Scandinavian Actuarial Journal, 2016 (2015), 741-762.  doi: 10.1080/03461238.2015.1015161. [17] J. Paulsen, Optimal dividend payouts for diffusions with solvency constraints, Finance and Stochastics, 7 (2003), 457-473.  doi: 10.1007/s007800200098. [18] J. Paulsen and H. Gjessing, Optimal choice of dividend barriers for a risk process with stochastic return on investments, Insurance: Mathematics and Economics, 20 (1997), 215-223.  doi: 10.1016/S0167-6687(97)00011-5. [19] V. Pestien and W. Sudderth, Continuous-time red and black: how to control a diffusion to a goal, Mathematics of Operations Research, 10 (1985), 599-611.  doi: 10.1287/moor.10.4.599. [20] H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68.  doi: 10.1080/034612301750077338. [21] M. Taksar and C. Markussen, Optimal dynamic reinsurance policies for large insurance portfolios, Finance and Stochastics, 7 (2003), 97-121.  doi: 10.1007/s007800200073. [22] N. Wang, Optimal investment for an insurer with exponential utility preference, Insurance: Mathematics and Economics, 40 (2007), 77-84.  doi: 10.1016/j.insmatheco.2006.02.008. [23] H. Yang and L. Zhang, Optimal investment for insurer with jump-diffusion risk process, Insurance: Mathematics and Economics, 37 (2005), 615-634.  doi: 10.1016/j.insmatheco.2005.06.009. [24] C. Yin and K. C. Yuen, Optimal dividend problems for a jump-diffusion model with capital injections and proportional transaction costs, Journal of Industrial and Management Optimization, 11 (2015), 1247-1262.  doi: 10.3934/jimo.2015.11.1247. [25] X. Zhang, M. Zhou and J. Guo, Optimal combinational quota-share and excess-of-loss reinsurance policies in a dynamic setting, Applied Stochastic Models in Business and Industry, 23 (2007), 63-71.  doi: 10.1002/asmb.637. [26] M. Zhou and K. Yuen, Optimal reinsurance and dividend for a diffusion model with capital injection: Variance premium principle, Economic Modelling, 29 (2012), 198-207.  doi: 10.1016/j.econmod.2011.09.007.
The minimal expected time and the associated optimal strategies for $\sigma=0.1$.
The minimal expected time and the associated optimal strategies for $\sigma=0.01$.
The minimal expected time and the associated optimal strategies for $b=0.03$.
The minimal expected time and the associated optimal strategies for $b=0.3$.
Expected time vs goal for $x=0.5$
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