Skip to main content
SpringerLink
Log in

Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes

  • Chapter
  • First Online:
Modern Problems in Insurance Mathematics

Part of the book series: EAA Series ((EAAS))

Abstract

We consider the problem of approximating the infinite time horizon ruin probabilities for discrete time risk processes. The approach is based on asymptotic results for non-linearly perturbed discrete time renewal equations. Under some moment conditions on the claim distributions, the approximations take the form of exponential asymptotic expansions with respect to the perturbation parameter. We show explicitly how the coefficients of these expansions can be computed as functions of the coefficients of the expansions of local characteristics for perturbed risk processes.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Blanchet, J., Zwart, B.: Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processors sharing queues. Math. Methods Oper. Res. 72(2), 311–326 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cossette, H., Landriault, D., Marceau, E.: Compound binomial risk model in a markovian environment. Insur. Math. Econ. 35, 425–443 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dickson, D.C.M., Egídio dos Reis, A.D., Waters, H.R.: Some stable algorithms in ruin theory and their applications. Astin Bull. 25(2), 153–175 (1995)

    Article  Google Scholar 

  4. Englund, E., Silvestrov, D.S.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers P., Kulldorff G., Portenko N., Silvestrov D. (eds.) Proceedings of the Second Scandinavian-Ukrainian Conference in Mathematical Statistics, vol. I, Umeå (1997) Theory stochastic Process, vol 3(19), no. 1–2, pp. 164–176 (1997)

    Google Scholar 

  5. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley Series in Probability and Statistics. Wiley, New York (1966, 1971)

    Google Scholar 

  6. Gerber, H.U.: Mathematical fun with the compound binomial process. Astin Bull. 18(2), 161–168 (1988)

    Article  Google Scholar 

  7. Grandell, J.: Aspects of Risk Theory. Probability and its Applications. Springer, New York (1991)

    Book  Google Scholar 

  8. Gyllenberg, M., Silvestrov, D.S.: Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol. 33, 35–70 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gyllenberg, M., Silvestrov, D.S.: Cramér-Lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econ. 26, 75–90 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (2008)

    Book  Google Scholar 

  11. Kartashov, M.V.: Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates. Theory Probab. Math. Stat. 78, 61–73 (2009)

    Article  MathSciNet  Google Scholar 

  12. Li, S., Lu, Y., Garrido, J.: A review of discrete-time risk models. Rev. R. Acad. Cien. Serie A. Mat. 103(2), 321–337 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ni, Y.: Perturbed renewal equations with multivariate non-polynomial perturbations. In: Frenkel I., Gertsbakh I., Khvatskin L., Laslo Z., Lisnianski A. (eds.) Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management, pp. 754–763. Beer Sheva, Israel (2010a)

    Google Scholar 

  14. Ni, Y.: Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations. J. Appl. Quant. Methods 5(3), 498–515 (2010b)

    Google Scholar 

  15. Petersson, M.: Quasi-stationary distributions for perturbed discrete time regenerative processes. Theory Probab. Math. Stat. 89 (2013, forthcoming)

    Google Scholar 

  16. Petersson, M., Silvestrov, D.: Asymptotic expansions for perturbed discrete time renewal equations and regenerative processes. Research Report 2012:12, Department of Mathematics, Stockholm University, 34 pp. (2012)

    Google Scholar 

  17. Shiu, E.S.W.: The probability of eventual ruin in the compound binomial model. Astin Bull. 19(2), 179–190 (1989)

    Article  Google Scholar 

  18. Silvestrov, D., Petersson, M.: Exponential expansions for perturbed discrete time renewal equations. In: Frenkel, I., Karagrigoriou, A., Lisnianski, A., Kleyner, A. (eds.) Applied Reliability Engineering and Risk Analysis: Probabilistic Models and Statistical Inference, pp. 349–362. Wiley, Chichester (2014)

    Google Scholar 

  19. Willmot, G.E.: Ruin probabilities in the compound binomial model. Insur. Math. Econ. 12, 133–142 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Stockholm University, Stockholm, Sweden

    Mikael Petersson

Authors
  1. Mikael Petersson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikael Petersson .

Editor information

Editors and Affiliations

  1. Department of Mathematics Professor of Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Dmitrii Silvestrov

  2. Department of Mathematics Professor in Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Anders Martin-Löf

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Petersson, M. (2014). Asymptotics of Ruin Probabilities for Perturbed Discrete Time Risk Processes. In: Silvestrov, D., Martin-Löf, A. (eds) Modern Problems in Insurance Mathematics. EAA Series. Springer, Cham. https://doi.org/10.1007/978-3-319-06653-0_7

Download citation

Publish with us

Policies and ethics

Search

Navigation