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RUIN PROBABILITIES UNDER AN OPTIMAL INVESTMENT AND PROPORTIONAL REINSURANCE POLICY IN A JUMP DIFFUSION RISK PROCESS

Part of: Mathematical economics Stochastic processes

Published online by Cambridge University Press:  09 March 2010

YIPING QIAN  and
XIANG LIN
YIPING QIAN
Affiliation:
School of Business, Central South University, Yuelu Mountain, Changsha 410083, Hunan, PR China
XIANG LIN*
Affiliation:
School of Mathematics, Central South University, No. 22 South Shaoshan Road, Changsha 410075, Hunan, PR China (email: xlin@csu.edu.cn)
*
✉For correspondence; e-mail: xlin@csu.edu.cn
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Abstract

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In this paper, we consider an insurance company whose surplus (reserve) is modeled by a jump diffusion risk process. The insurance company can invest part of its surplus in n risky assets and purchase proportional reinsurance for claims. Our main goal is to find an optimal investment and proportional reinsurance policy which minimizes the ruin probability. We apply stochastic control theory to solve this problem. We obtain the closed form expression for the minimal ruin probability, optimal investment and proportional reinsurance policy. We find that the minimal ruin probability satisfies the Lundberg equality. We also investigate the effects of the diffusion volatility parameter, the market price of risk and the correlation coefficient on the minimal ruin probability, optimal investment and proportional reinsurance policy through numerical calculations.

Keywords

ruin probabilityproportional reinsuranceoptimal investment policyLundberg’s equalityHamilton–Jacobi–Bellman equation

MSC classification

Secondary: 60G99: None of the above, but in this section 91B30: Risk theory, insurance
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 1 , July 2009 , pp. 34 - 48
Copyright
Copyright © Australian Mathematical Society 2010

References

[1]Browne, S., “Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin”, Math. Methods Oper. Res. 20 (1995) 937–957.CrossRefGoogle Scholar
[2]Browne, S., “Survival and growth with liability: optimal portfolio strategies in continuous time”, Math. Methods Oper. Res. 22 (1997) 468–493.CrossRefGoogle Scholar
[3]Browne, S., “Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark”, Finance Stoch. 3 (1999) 275–294.CrossRefGoogle Scholar
[4]Dufresne, F. and Gerber, H. U., “Risk theory for the compound Poisson process that is perturbed by diffusion”, Insurance Math. Econom. 10 (1991) 51–59.Google Scholar
[5]Gaier, J., Grandits, P. and Schachermayer, W., “Asymptotic ruin probabilities and optimal investment”, Ann. Appl. Probab. 13 (2003) 1054–1076.Google Scholar
[6]Hald, M. and Schmidli, H., “On the maximization of the adjustment coefficient under proportional reinsurance”, Astin Bull. 34 (2004) 75–83.Google Scholar
[7]Hipp, C. and Plum, M., “Optimal investment for insurers”, Insurance Math. Econom. 27 (2000) 215–228.CrossRefGoogle Scholar
[8]Hipp, C. and Plum, M., “Optimal investment for investors with state dependent income, and for insurers”, Finance Stoch. 7 (2003) 299–321.CrossRefGoogle Scholar
[9]Liang, Z. B., “Optimal proportional reinsurance for controlled risk process which is perturbed by diffusion”, Acta Math. Appl. Sin. Engl. Ser. 23 (2007) 477–488.CrossRefGoogle Scholar
[10]Liang, Z. B. and Guo, J. Y., “Optimal proportional reinsurance and ruin probability”, Stochastic Models 23 (2007) 333–350.CrossRefGoogle Scholar
[11]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.CrossRefGoogle Scholar
[12]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.CrossRefGoogle Scholar
[13]Liu, C. S. and Yang, H., “Optimal investment for an insurer to minimize its probability of ruin”, N. Am. Actuar. J. 8 (2004) 11–31.Google Scholar
[14]Luo, S. Z., Taksar, M. and Tsoi, A., “On reinsurance and investment for large insurance portfolios”, Insurance Math. Econom. 42 (2008) 434–444.CrossRefGoogle Scholar
[15]Øksendal, B. and Sulem, A., Applied stochastic control in jump diffusions (Springer, Berlin, 2005).Google Scholar
[16]Promislow, S. D. and Young, V. R., “Minimizing the probability of ruin when claims follow Brownian motion with drift”, N. Am. Actuar. J. 9(3) (2005) 109–128.Google Scholar
[17]Schmidli, H., “Optimal proportional reinsurance policies in a dynamic setting”, Scand. Actuar. J. 1 (2002) 55–68.Google Scholar
[18]Schmidli, H., “On minimizing the ruin probability by investment and reinsurance”, Ann. Appl. Probab. 12 (2002) 890–907.CrossRefGoogle Scholar
[19]Schmidli, H., Stochastic control in insurance (Springer, London, 2008).Google Scholar
[20]Taksar, M. I. and Markussen, C., “Optimal dynamic reinsurance policies for large insurance portfolios”, Finance Stoch. 7 (2003) 97–121.Google Scholar
[21]Wang, G. and Wu, R., “Some distributions for classical risk process that is perturbed by diffusion”, Insurance Math. Econom. 26 (2000) 15–24.Google Scholar
[22]Yang, H. and Zhang, L., “Optimal investment for insurer with jump-diffusion risk process”, Insurance Math. Econom. 37 (2005) 615–634.Google Scholar