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OPTIMAL INVESTMENT AND REINSURANCE IN A JUMP DIFFUSION RISK MODEL

Published online by Cambridge University Press:  14 October 2011

XIANG LIN  and
PENG YANG
XIANG LIN*
Affiliation:
School of Mathematical Science and Computing Technology, Central South University, No. 22 South Shaoshan Road, Changsha 410075, Hunan, PR China (email: xlin@csu.edu.cn, yangpeng511@163.com)
PENG YANG
Affiliation:
School of Mathematical Science and Computing Technology, Central South University, No. 22 South Shaoshan Road, Changsha 410075, Hunan, PR China (email: xlin@csu.edu.cn, yangpeng511@163.com)
*
✉For correspondence; e-mail: xlin@csu.edu.cn
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Abstract

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We consider an insurance company whose surplus is governed by a jump diffusion risk process. The insurance company can purchase proportional reinsurance for claims and invest its surplus in a risk-free asset and a risky asset whose return follows a jump diffusion process. Our main goal is to find an optimal investment and proportional reinsurance policy which maximizes the expected exponential utility of the terminal wealth. By solving the corresponding Hamilton–Jacobi–Bellman equation, closed-form solutions for the value function as well as the optimal investment and proportional reinsurance policy are obtained. We also discuss the effects of parameters on the optimal investment and proportional reinsurance policy by numerical calculations.

Keywords

jump diffusion risk modelproportional reinsuranceinvestmentcompound Poisson processexponential utilityHamilton–Jacobi–Bellman equation
Type
Research Article
Information
The ANZIAM Journal , Volume 52 , Issue 3 , January 2011 , pp. 250 - 262
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Asmussen, S. and Taksar, M., “Controlled diffusion models for optimal dividend pay-out”, Insurance Math. Econ. 20 (1997) 1–15; doi:10.1016/S0167-6687(96)00017-0.CrossRefGoogle Scholar
[2]Bai, L. and Guo, J., “Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint”, Insurance Math. Econ. 42 (2008) 968–975; doi:10.1016/j.insmatheco.2007.11.002.CrossRefGoogle Scholar
[3]Browne, S., “Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin”, Math. Oper. Res. 20 (1995) 937–958; doi:10.1287/moor.20.4.937.CrossRefGoogle Scholar
[4]Browne, S., “Survival and growth with a liability: optimal portfolio strategies in continuous time”, Math. Oper. Res. 22 (1997) 468–493; doi:10.1287/moor.22.2.468.CrossRefGoogle Scholar
[5]Browne, S., “Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark”, Finance Stoch. 3 (1999) 275–294; doi:10.1007/s007800050063.CrossRefGoogle Scholar
[6]Cramer, H., Collective risk theory, Jubilee Volume (Skandia Insurance Company, Stockholm, 1955).Google Scholar
[7]Dufresne, F. and Gerber, H. U., “Risk theory for the compound Poisson process that is perturbed by diffusion”, Insurance Math. Econ. 10 (1991) 51–59; doi:10.1016/0167-6687(91)90023-Q.CrossRefGoogle Scholar
[8]Fleming, W. H. and Soner, H. M., Controlled Markov processes and viscosity solutions (Springer, Berlin–New York, 1993).Google Scholar
[9]Gaier, J., Grandits, P. and Schachermayer, W., “Asymptotic ruin probabilities and optimal investment”, Ann. Appl. Probab. 13 (2003) 1054–1076.CrossRefGoogle Scholar
[10]Gerber, H., An introduction to mathematical risk theory, Volume 8 of S. S. Huebner Foundation Monograph Series (ed. Huebner, S. S.), (Foundation for Insurance Education, Philadelphia, PA, 1979).Google Scholar
[11]Goovaerts, M. J., Kass, R., van Heerwarrden, A. E. and Bauwelinckx, T., Effective actuarial methods (North-Holland, Amsterdam, 1990).Google Scholar
[12]Hald, M. and Schmidli, H., “On the maximization of the adjustment coefficient under proportional reinsurance”, ASTIN Bull. 34 (2004) 75–83; doi:10.2143/AST.34.1.504955.CrossRefGoogle Scholar
[13]Hipp, C. and Plum, M., “Optimal investment for insurers”, Insurance Math. Econ. 27 (2000) 215–228; doi:10.1016/S0167-6687(00)00049-4.CrossRefGoogle Scholar
[14]Hipp, C. and Plum, M., “Optimal investment for investors with state dependent income, and for insurers”, Finance Stoch. 7 (2003) 299–321; doi:10.1007/s007800200095.CrossRefGoogle Scholar
[15]Højgaard, B. and Taksar, M., “Optimal proportional reinsurance policies for diffusion models”, Scand. Actuar. J. (1998) 166–180.CrossRefGoogle Scholar
[16]Højgaard, B. and Taksar, M., “Controlling risk exposure and dividend pay-out schemes: insurance company example”, Math. Finance 9 (1999) 153–182; doi:10.1111/1467-9965.00066.CrossRefGoogle Scholar
[17]Irgens, C. and Paulsen, J., “Optimal control of risk exposure, reinsurance and investments for insurance portfolios”, Insurance Math. Econ. 35 (2004) 21–51; doi:10.1016/j.insmatheco.2004.04.004.CrossRefGoogle Scholar
[18]Jeanblanc-Picqué, M. and Shiryaev, A. N., “Optimization of the flow of dividends”, Russian Math. Surveys 50 (1995) 257–277; doi:10.1070/RM1995v050n02ABEH002054.CrossRefGoogle Scholar
[19]Kou, S. G. and Wang, H., “First passage times of a jump diffusion process”, Adv. Appl. Probab. 35 (2003) 504–531; doi:10.1239/aap/1051201658.CrossRefGoogle Scholar
[20]Liang, Z. B. and Guo, J. Y., “Optimal proportional reinsurance and ruin probability”, Stoch. Models 23 (2007) 333–350; doi:10.1080/15326340701300894.CrossRefGoogle Scholar
[21]Liang, Z. B. and Guo, J. Y., “Upper bound for ruin probabilities under optimal investment and proportional reinsurance”, Appl. Stoch. Models Bus. Ind. 24 (2008) 109–128; doi:10.1002/asmb.694.CrossRefGoogle Scholar
[22]Lin, X., “Ruin theory for classical risk process that is perturbed by diffusion with risky investments”, Appl. Stoch. Models Bus. Ind. 25 (2009) 33–44; doi:10.1002/asmb.719.CrossRefGoogle Scholar
[23]Liu, C. S. and Yang, H. L., “Optimal investment for an insurer to minimize its probability of ruin”, N. Am. Actuar. J. 8 (2004) 11–31.CrossRefGoogle Scholar
[24]Luo, S. Z., Taksar, M. and Tsoi, A., “On reinsurance and investment for large insurance portfolios”, Insurance Math. Econ. 42 (2008) 434–444; doi:10.1016/j.insmatheco.2007.04.002.CrossRefGoogle Scholar
[25]Promislow, S. D. and Young, V. R., “Minimizing the probability of ruin when claims follow Brownian motion with drift”, N. Am. Actuar. J. 9 (2005) 109–128.Google Scholar
[26]Qian, Y. P. and Lin, X., “Ruin probabilities under an optimal investment and proportional reinsurance policy in a jump diffusion risk process”, ANZIAM J. 51 (2009) 34–48; doi:10.1017/S144618110900042X.CrossRefGoogle Scholar
[27]Schmidli, H., “Optimal proportional reinsurance policies in a dynamic setting”, Scand. Actuar. J. (2001) 55–68; doi:10.1080/034612301750077338.CrossRefGoogle Scholar
[28]Schmidli, H., “On minimizing the ruin probability by investment and reinsurance”, Ann. Appl. Probab. 12 (2002) 890–907; doi:10.1214/aoap/1031863173.CrossRefGoogle Scholar
[29]Taksar, M. and Markussen, C., “Optimal dynamic reinsurance policies for large insurance portfolios”, Finance Stoch. 7 (2003) 97–121; doi:10.1007/s007800200073.CrossRefGoogle Scholar
[30]Yang, H. L. and Zhang, L. H., “Optimal investment for insurer with jump-diffusion risk process”, Insurance Math. Econ. 37 (2005) 615–634; doi:10.1016/j.insmatheco.2005.06.009.CrossRefGoogle Scholar