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




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