American Institute of Mathematical Sciences

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March  2019, 9(1): 59-76. doi: 10.3934/mcrf.2019003

Robust optimal investment and reinsurance of an insurer under Jump-diffusion models

Xin Zhang 1,, , Hui Meng 2, , Jie Xiong 3, and  Yang Shen 4,

1. 

School of Mathematics, Southeast University, Nanjing, Jiangsu Province, 211189, China
2. 

China Institute for Actuarial Science, Central University of Finance and Economics, Beijing 100081, China
3. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong Province, 518055, China
4. 

Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

* Corresponding author: Xin Zhang

Received  August 2017 Revised  April 2018 Published  August 2018

Fund Project: X. Zhang is supported by the National Natural Science Foundation of China (grant nos. 11771079, 11371020). H. Meng is supported by the National Natural Science Foundation of China (grant no. 11771465), the Program for Innovation Research in Central University of Finance and Economics, and the 111 Project (grant no. B17050). J. Xiong is supported by Southern University of Science and Technology startup fund (grant No. 28/Y01286120). Y. Shen is supported by the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2016-05677).

This paper studies a robust optimal investment and reinsurance problem under model uncertainty. The insurer's risk process is modeled by a general jump process generated by a marked point process. By transferring a proportion of insurance risk to a reinsurance company and investing the surplus into the financial market with a bond and a share index, the insurance company aims to maximize the minimal expected terminal wealth with a penalty. By using the dynamic programming, we formulate the robust optimal investment and reinsurance problem into a two-person, zero-sum, stochastic differential game between the investor and the market. Closed-form solutions for the case of the quadratic penalty function are derived in our paper.

Keywords: Robust control, proportional reinsurance, stochastic differential game, HJBI equation, jump-diffusion model.
Mathematics Subject Classification: Primary: 93E20, 91G80; Secondary: 91A15.
Citation: Xin Zhang, Hui Meng, Jie Xiong, Yang Shen. Robust optimal investment and reinsurance of an insurer under Jump-diffusion models. Mathematical Control and Related Fields, 2019, 9 (1) : 59-76. doi: 10.3934/mcrf.2019003
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]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1.

[4]

N. Branger and L.S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, Journal of Banking and Finance, 37 (2013), 5036-5047.  doi: 10.1016/j.jbankfin.2013.08.023.

[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]

S. Browne, Survival and growth with a liability: Optimal portfolio strategies in continuous time, Mathematics of Operations Research, 22 (1997), 468-493.  doi: 10.1287/moor.22.2.468.

[7]

S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3 (1999), 275-294.  doi: 10.1007/s007800050063.

[8]

A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330.  doi: 10.1016/S0167-6687(00)00055-X.

[9]

Á. Cartea and S. Jaimungal, Risk metrics and fine tuning of high-frequency trading strategies, Mathematical Finance, 25 (2015), 576-611.  doi: 10.1111/mafi.12023.

[10]

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.

[11]

R. Cont, Model uncertanity and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2006), 519-547.  doi: 10.1111/j.1467-9965.2006.00281.x.

[12]

J. Dupačová and J. Polívka, Asset-liability management for czech pension funds using stochastic programming, Annals of Operations Research, 165 (2009), 5-28.  doi: 10.1007/s10479-008-0358-6.

[13] R. Elliott, Stochastic Calculus and Applications, Springer Verlag, New York, 1982. 
[14] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. 
[15]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.

[16]

C. Hipp and M. Taksar, Stochastic control for optimal new business, Insurance: Mathematics and Economics, 26 (2000), 185-192.  doi: 10.1016/S0167-6687(99)00052-9.

[17]

B. Højgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.

[18]

Z. Liang, Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Mathematica Sinica, Chinese Series, 51 (2008), 1195-1204. 

[19]

X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the cev model, North American Actuarial Journal, 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.

[20]

C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.

[21]

S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002.

[22]

F. Maccheroni, M. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498.  doi: 10.1111/j.1468-0262.2006.00716.x.

[23]

P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003.

[24]

S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337.  doi: 10.1080/17442500701655408.

[25]

H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies, Econocmic Modelling, 28 (2011), 211-218.  doi: 10.1016/j.econmod.2010.09.009.

[26]

H. Meng, T. Siu and H. Yang, Optimal dividends with debts and nonlinear insurance risk processes, Insurance: Mathematics and Economics, 53 (2013), 110-121.  doi: 10.1016/j.insmatheco.2013.04.008.

[27]

C. Moallemi and M. Sağlam, Dynamic portfolio choice with linear rebalancing rules, Journal of Financial and Quantitative Analysis, 52 (2017), 1247-1278. 

[28]

National Association of Insurance Commissioners, Capital Markets Special Report: U. S. Insurance Industry Cash and Invested Assets at Year-End 2016. Available at http://www.naic.org/capital_markets_archive/170824.htm(2007).

[29]

B. Øksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Mathematical Finance, 19 (2009), 619-637.  doi: 10.1111/j.1467-9965.2009.00382.x.

[30]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68.  doi: 10.1080/034612301750077338.

[31]

H. Schmidli, On minimising the ruin probability by investment and reinsurance, Annal of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[32]

Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.

[33]

M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001.

[34]

Z. Wen, X. Wu and Y. Zhou, Dynamic ordering policies under partial trade credit financing, in Service Systems and Service Management (ICSSSM), 2014 11th International Conference on, IEEE, 2014, 1-6. doi: 10.1109/ICSSSM.2014.6874077.

[35]

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.

[36]

B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scandinavian Actuarial Journal, 2015 (2015), 725-751.  doi: 10.1080/03461238.2014.883085.

[37]

C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.  doi: 10.1016/j.insmatheco.2013.09.019.

[38]

C. Yin, Y. Wen and Y. Zhao, On the optimal dividend problem for a spectrally positive levy process, ASTIN Bulletin, 44 (2014), 635-651.  doi: 10.1017/asb.2014.12.

[39]

V.R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126.  doi: 10.1080/10920277.2004.10596174.

[40]

J. Zhang and Q. Xiao, Optimal investment of a time-dependent renewal risk model with stochastic return, Journal of Inequalities and Applications, 2015 (2015), 12pp.  doi: 10.1186/s13660-015-0707-3.

[41]

X. Zhang and T. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Mathematics and Economics, 45 (2009), 81-88.  doi: 10.1016/j.insmatheco.2009.04.001.

[42]

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: Mathematics and Economics, 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001.

[43]

X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a cev model, Insurance: Mathematics and Economics, 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008.

[44]

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]

N. Bäuerle, Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62 (2005), 159-165.  doi: 10.1007/s00186-005-0446-1.

[4]

N. Branger and L.S. Larsen, Robust portfolio choice with uncertainty about jump and diffusion risk, Journal of Banking and Finance, 37 (2013), 5036-5047.  doi: 10.1016/j.jbankfin.2013.08.023.

[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]

S. Browne, Survival and growth with a liability: Optimal portfolio strategies in continuous time, Mathematics of Operations Research, 22 (1997), 468-493.  doi: 10.1287/moor.22.2.468.

[7]

S. Browne, Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark, Finance and Stochastics, 3 (1999), 275-294.  doi: 10.1007/s007800050063.

[8]

A. Cairns, A discussion of parameter and model uncertainty in insurance, Insurance: Mathematics and Economics, 27 (2000), 313-330.  doi: 10.1016/S0167-6687(00)00055-X.

[9]

Á. Cartea and S. Jaimungal, Risk metrics and fine tuning of high-frequency trading strategies, Mathematical Finance, 25 (2015), 576-611.  doi: 10.1111/mafi.12023.

[10]

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.

[11]

R. Cont, Model uncertanity and its impact on the pricing of derivative instruments, Mathematical Finance, 16 (2006), 519-547.  doi: 10.1111/j.1467-9965.2006.00281.x.

[12]

J. Dupačová and J. Polívka, Asset-liability management for czech pension funds using stochastic programming, Annals of Operations Research, 165 (2009), 5-28.  doi: 10.1007/s10479-008-0358-6.

[13] R. Elliott, Stochastic Calculus and Applications, Springer Verlag, New York, 1982. 
[14] W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. 
[15]

C. Hipp and M. Plum, Optimal investment for insurers, Insurance: Mathematics and Economics, 27 (2000), 215-228.  doi: 10.1016/S0167-6687(00)00049-4.

[16]

C. Hipp and M. Taksar, Stochastic control for optimal new business, Insurance: Mathematics and Economics, 26 (2000), 185-192.  doi: 10.1016/S0167-6687(99)00052-9.

[17]

B. Højgaard and M. Taksar, Controlling risk exposure and dividends payout schemes: Insurance company example, Mathematical Finance, 9 (1999), 153-182.  doi: 10.1111/1467-9965.00066.

[18]

Z. Liang, Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Mathematica Sinica, Chinese Series, 51 (2008), 1195-1204. 

[19]

X. Lin and Y. Li, Optimal reinsurance and investment for a jump diffusion risk process under the cev model, North American Actuarial Journal, 15 (2011), 417-431.  doi: 10.1080/10920277.2011.10597628.

[20]

C. Liu and H. Yang, Optimal investment for an insurer to minimize its probability of ruin, North American Actuarial Journal, 8 (2004), 11-31.  doi: 10.1080/10920277.2004.10596134.

[21]

S. Luo, M. Taksar and A. Tsoi, On reinsurance and investment for large insurance portfolios, Insurance: Mathematics and Economics, 42 (2008), 434-444.  doi: 10.1016/j.insmatheco.2007.04.002.

[22]

F. Maccheroni, M. Marinacci and A. Rustichini, Ambiguity aversion, robustness, and the variational representation of preferences, Econometrica, 74 (2006), 1447-1498.  doi: 10.1111/j.1468-0262.2006.00716.x.

[23]

P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983.  doi: 10.1093/rfs/hhh003.

[24]

S. Mataramvura and B. Øksendal, Risk minimizing portfolios and HJBI equations for stochastic differential games, Stochastics An International Journal of Probability and Stochastic Processes, 80 (2008), 317-337.  doi: 10.1080/17442500701655408.

[25]

H. Meng and T. Siu, On optimal reinsurance, dividend and reinvestment strategies, Econocmic Modelling, 28 (2011), 211-218.  doi: 10.1016/j.econmod.2010.09.009.

[26]

H. Meng, T. Siu and H. Yang, Optimal dividends with debts and nonlinear insurance risk processes, Insurance: Mathematics and Economics, 53 (2013), 110-121.  doi: 10.1016/j.insmatheco.2013.04.008.

[27]

C. Moallemi and M. Sağlam, Dynamic portfolio choice with linear rebalancing rules, Journal of Financial and Quantitative Analysis, 52 (2017), 1247-1278. 

[28]

National Association of Insurance Commissioners, Capital Markets Special Report: U. S. Insurance Industry Cash and Invested Assets at Year-End 2016. Available at http://www.naic.org/capital_markets_archive/170824.htm(2007).

[29]

B. Øksendal and A. Sulem, Risk indifference pricing in jump diffusion markets, Mathematical Finance, 19 (2009), 619-637.  doi: 10.1111/j.1467-9965.2009.00382.x.

[30]

H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68.  doi: 10.1080/034612301750077338.

[31]

H. Schmidli, On minimising the ruin probability by investment and reinsurance, Annal of Applied Probability, 12 (2002), 890-907.  doi: 10.1214/aoap/1031863173.

[32]

Z. Sun, X. Zheng and X. Zhang, Robust optimal investment and reinsurance of an insurer under variance premium principle and default risk, Journal of Mathematical Analysis and Applications, 446 (2017), 1666-1686.  doi: 10.1016/j.jmaa.2016.09.053.

[33]

M. Taksar, Optimal risk and dividend distribution control models for an insurance company, Mathematical Methods of Operations Research, 51 (2000), 1-42.  doi: 10.1007/s001860050001.

[34]

Z. Wen, X. Wu and Y. Zhou, Dynamic ordering policies under partial trade credit financing, in Service Systems and Service Management (ICSSSM), 2014 11th International Conference on, IEEE, 2014, 1-6. doi: 10.1109/ICSSSM.2014.6874077.

[35]

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.

[36]

B. Yi, F. Viens, Z. Li and Y. Zeng, Robust optimal strategies for an insurer with reinsurance and investment under benchmark and mean-variance criteria, Scandinavian Actuarial Journal, 2015 (2015), 725-751.  doi: 10.1080/03461238.2014.883085.

[37]

C. Yin and Y. Wen, Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance: Mathematics and Economics, 53 (2013), 769-773.  doi: 10.1016/j.insmatheco.2013.09.019.

[38]

C. Yin, Y. Wen and Y. Zhao, On the optimal dividend problem for a spectrally positive levy process, ASTIN Bulletin, 44 (2014), 635-651.  doi: 10.1017/asb.2014.12.

[39]

V.R. Young, Optimal investment strategy to minimize the probability of lifetime ruin, North American Actuarial Journal, 8 (2004), 105-126.  doi: 10.1080/10920277.2004.10596174.

[40]

J. Zhang and Q. Xiao, Optimal investment of a time-dependent renewal risk model with stochastic return, Journal of Inequalities and Applications, 2015 (2015), 12pp.  doi: 10.1186/s13660-015-0707-3.

[41]

X. Zhang and T. Siu, Optimal investment and reinsurance of an insurer with model uncertainty, Insurance Mathematics and Economics, 45 (2009), 81-88.  doi: 10.1016/j.insmatheco.2009.04.001.

[42]

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: Mathematics and Economics, 67 (2016), 125-132.  doi: 10.1016/j.insmatheco.2016.01.001.

[43]

X. Zheng, J. Zhou and Z. Sun, Robust optimal portfolio and proportional reinsurance for an insurer under a cev model, Insurance: Mathematics and Economics, 67 (2016), 77-87.  doi: 10.1016/j.insmatheco.2015.12.008.

[44]

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.

Figure 1.  Effects of $\lambda$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
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Figure 2.  Effects of $q$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
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Figure 3.  Effects of $\beta_1$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
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Figure 4.  Effects of $\beta_2$ on the optimal strategies $({\hat \pi}_1, {\hat \pi}_2)$ with different $\zeta$
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