Skip to main content
SpringerLink
Log in

Optimal Proportional Reinsurance for Controlled Risk Process which is Perturbed by Diffusion

  • Original Papers
  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we study optimal proportional reinsurance policy of an insurer with a risk process which is perturbed by a diffusion. We derive closed-form expressions for the policy and the value function, which are optimal in the sense of maximizing the expected utility in the jump-diffusion framework. We also obtain explicit expressions for the policy and the value function, which are optimal in the sense of maximizing the expected utility or maximizing the survival probability in the diffusion approximation case. Some numerical examples are presented, which show the impact of model parameters on the policy. We also compare the results under the different criteria and different cases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asmussen, S., Taksar, M. Controlled diffusion models for optimal dividend pay-out. Insurance: Math. Econom., 20:1–15 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  2. Browne, S. Optimal investment policies for a firm with random risk process:exponential utility and minimizing the probability of ruin. Math.Oper.Res, 20:937–958 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cont, R., Tankov, P. Financial modelling with jump processes. Chapman and Hall/CRC Financial Mathematics Series, 2003

  4. Fleming, W., Sonor, H. Controlled Markov Processes and viscosity solutions. Springer-Verlag, New York, 1993

  5. Gerber, H. An introduction to mathematical risk theory. Huebner Foundation Monograph. No. 8, 1979

  6. Gerber, H., Shiu, E. Optimal dividends:analysis with Brownian motion. North American Actuarial Journal, 8(1):1–20 (2004)

    MATH  MathSciNet  Google Scholar 

  7. Grandell, J. Aspects of risk theory. Springer-Verlag, New York, 1991

  8. Hipp, C., Plum, M. Optimal investment for insurers. Insurance: Math. Econom., 27:215–228 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hipp, C., Plum, M. Optimal investment for investors with state dependent income, and for insurers. Finance and Stochastics, 7:299–321 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hφjgaard, B., Taksar, H. Optimal proportional reinsurance policies for diffusion models. Scand. Actuarial. J., 166-180 (1998)

  11. Karatzas, I. Optimization problems in the theory of continuous trading. SIAM J. Control. and Optim., 7:1221–1259 (1989)

    Article  MathSciNet  Google Scholar 

  12. Liu, C., Yang, H. Optimal investment for a insurer to minimize its probability of ruin. North American Acruarial Journal, 8(2):11–31 (2004)

    MATH  Google Scholar 

  13. Merton, R. Optimum consumption and protfolio rules in a continuous-time model. J. Econom. Theory, 3:373–413 (1971)

    Article  MathSciNet  Google Scholar 

  14. Merton, R. Continuous-Time Finance, Blackwell, Massachusetts, 1990

  15. Schmidli, H. Optimal proportional reinsurance policies in a dynamic setting. Scand. Actuarial. J., 55–68 (2001)

  16. Schmidli, H. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab , 12:890–907 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Yang, H., Zhang, L. Optimal investment for insurer with jump-diffusion risk process. Insurance: Math. Econom, 37:615–634 (2005)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Nanjing Normal University, Nanjing, 210097, China

    Zhi-bin Liang

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Zhi-bin Liang

Authors
  1. Zhi-bin Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-bin Liang.

Additional information

Supported by the National Natural Science Foundation of China (No. 10571092)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liang, Zb. Optimal Proportional Reinsurance for Controlled Risk Process which is Perturbed by Diffusion. Acta Mathematicae Applicatae Sinica, English Series 23, 477–488 (2007). https://doi.org/10.1007/s10255-007-0387-y

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-007-0387-y

Keywords

2000 MR Subject Classification

Search

Navigation