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

Xin Zhang; Hui Meng; Jie Xiong; Yang Shen

doi:10.3934/mcrf.2019003

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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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	R. Elliott, Stochastic Calculus and Applications, Springer Verlag, New York, 1982. Google Scholar
[14]	W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	Z. Liang, Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Mathematica Sinica, Chinese Series, 51 (2008), 1195-1204. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983. doi: 10.1093/rfs/hhh003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	C. Moallemi and M. Sağlam, Dynamic portfolio choice with linear rebalancing rules, Journal of Financial and Quantitative Analysis, 52 (2017), 1247-1278. Google Scholar
[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). Google Scholar
[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. Google Scholar
[30]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68. doi: 10.1080/034612301750077338. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	R. Elliott, Stochastic Calculus and Applications, Springer Verlag, New York, 1982. Google Scholar
[14]	W. Fleming and H. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 2006. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	Z. Liang, Optimal investment and reinsurance for the jump-diffusion surplus processes, Acta Mathematica Sinica, Chinese Series, 51 (2008), 1195-1204. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	P. J. Maenhout, Robust portfolio rules and asset pricing, Review of Financial Studies, 17 (2004), 951-983. doi: 10.1093/rfs/hhh003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[27]	C. Moallemi and M. Sağlam, Dynamic portfolio choice with linear rebalancing rules, Journal of Financial and Quantitative Analysis, 52 (2017), 1247-1278. Google Scholar
[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). Google Scholar
[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. Google Scholar
[30]	H. Schmidli, Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 2001 (2001), 55-68. doi: 10.1080/034612301750077338. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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