Skip to main content
SpringerLink
Log in

A review of discrete-time risk models

Una revista de modelos de riesgo en tiempo discreto

  • Published:
RACSAM - Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

In this paper, we present a review of results for discrete-time risk models, including the compound binomial risk model and some of its extensions. While most theoretical risk models use the concept of time continuity, the practical reality is discrete. For instance, recursive formulas for discretetime models can be obtained without assuming a claim severity distribution and are readily programmable in practice. Hence themodels, techniques used, and results reviewed here for discrete-time risk models are of independent scientific interest. Yet, results for discrete-time risk models can give, in addition, a simpler understanding of their continuous-time analogue. For example, these results can serve as approximations or bounds for the corresponding results in continuous-time models. This paper will serve as a detailed reference for the study of discrete-time risk models.

Resumen

En este artículo hacemos un repaso de los resultados para modelos de riesgo en tiempo discreto, incluyendo el modelo de riesgo binomial-compuesto, así como algunas de sus extensiones. Aunque gran parte de los modelos teóricos de riesgo se basen en el concepto de continuidad del tiempo, la realidad práctica es en sí discreta. Por ejemplo, en la práctica actuarial se programan fórmulas recursivas para modelos en tiempo discreto, sin necesidad de suponer una distribución de pérdidas conocida. Con lo cual estos modelos, las técnicas y los resultados que listamos para modelos de riesgo en tiempo discreto, generan un cierto interés científico propio. Pero más allá de sus aplicaciones directas, estos resultados para modelos en tiempo discreto también proporcionan un camino más simple hacia los modelos de riesgo análogos en tiempo continuo. Por ejemplo, los resultados en tiempo discreto pueden servir de aproximaciones o de cotas para sus resultados correspondientes en tiempo continuo. El propósito de este artículo es que pueda servir de referencia detallada para el estudio de modelos de riesgo en tiempo discreto.

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. Andersen, E. S., (1957). On the collective theory of risk in case of contagion between claims,Bulletin of the Institute of Mathematics and its Applications,12, 275–279.

    Google Scholar 

  2. Asmussen, S., (1989). Ruin theory in a Markovian environment,Scand. Actuar. J.,1989, 69–100.

    MATH  MathSciNet  Google Scholar 

  3. Asmussen, S., (1992). Phase-type representations in random walk and queuing problems,Ann. Probab.,20, 772–789.

    Article  MATH  MathSciNet  Google Scholar 

  4. Asmussen, S., (2000).Ruin probabilities, World Scientific, Singapore.

    Book  Google Scholar 

  5. Bobbio, A., Horváth, A., Scarpa, M. andTeleK, M., (2003). Acyclic discrete phase type distributions: properties and a parameter estimation algorithm,Performance Evaluation,54, 1–32.

    Article  Google Scholar 

  6. Cheng, S., Gerber, H. U. andShiu, E. S. W., (2000). Discounted probabilities and ruin theory in the compound binomial model,Insurance Math. Econom.,26, 239–250.

    Article  MATH  MathSciNet  Google Scholar 

  7. Cossette, H., Landriault, D. andMarceau, E., (2003). Ruin probabilities in the compound Markov binomial model,Scand. Actuar. J.,2003(4), 301–323.

    Article  MATH  MathSciNet  Google Scholar 

  8. Cossette, H., Landriault, D. andMarceau, E., (2004). Compound binomial risk model in a Markovian environment,Insurance Math. Econom.,35, 425–443.

    Article  MATH  MathSciNet  Google Scholar 

  9. Cossette, H., Landriault, D. andMarceau, E., (2006). Ruin probabilities in the discrete time renewal risk model,Insurance Math. Econom.,38, 309–323.

    Article  MATH  MathSciNet  Google Scholar 

  10. Dickson, D. C. M., (1992). On the distribution of surplus prior to ruin,Insurance Math. Econom.,11, 191–207.

    Article  MATH  MathSciNet  Google Scholar 

  11. Dickson, D. C. M., (1994). Some comments on the compound binomial model,Astin Bull.,24, 33–45.

    Article  Google Scholar 

  12. Dickson, D. C. M., Egidio Reis, A. D. andWaters, H. R., (1995). Some stable algorithms in ruin theory and their applications,Astin Bull.,25, 153–175.

    Article  Google Scholar 

  13. Dickson, D. C. M. andHipp, C., (2001). On the time to ruin for Erlang (2) risk process.Insurance Math. Econom.,29, 333–344.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  15. Gerber, H. U. andShiu, E. S. W., (1998). On the time value of ruin,N. Am. Actuar. J.,2(1), 48–78.

    MATH  MathSciNet  Google Scholar 

  16. Gerber, H. U. andShiu, E. S. W., (2005). The time value of ruin in a Sparre Andersen model,N. Am. Actuar. J.,9(2), 49–69.

    MATH  MathSciNet  Google Scholar 

  17. Landriault, D., (2008). On a generalization of the expected discounted penalty function in a discrete-time insurance risk model,Appl. Stoch. Models Bus. Ind.,24, 525–539.

    Article  MathSciNet  Google Scholar 

  18. Latouche, G. andRamaswami, V., (1999).Introduction to matrix analytic methods in stochastic modeling. ASA-SIAM, Philadelphia.

    MATH  Google Scholar 

  19. Li, S., (2005). On a class of discrete-time renewal risk models,Scand. Actuar. J.,2005(4), 241–260.

    Article  MATH  Google Scholar 

  20. Li, S., (2005). Distributions of the surplus before ruin, the deficit at ruin and the claim causing ruin in a class of discrete time risk models,Scand. Actuar. J.,2005(4), 271–284.

    Article  MATH  Google Scholar 

  21. Li, S. and Garrido, J., (2002). On the time value of ruin in the discrete time risk model,Working paper 02–18, Business Economics,University Carlos III of Madrid, 1-28. http://www.uc3m.es/portal/page/portal/dpto economia empresa/investigacion/ workingpaper/workingpapers02/wb021812.pdf

  22. Li, S. andGarrido, J., (2004). On ruin for Erlang(n) risk process,Insurance Math. Econom.,34, 391–408.

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, S. andGarrido, J., (2005). On a general class of renewal risk process: Analysis of the Gerber-Shiu penalty function,Adv. in Appl. Probab.,37, 836–856.

    Article  MATH  MathSciNet  Google Scholar 

  24. Lin, X. S. andWillmot, G. E., (1999). Analysis of a defective renewal arising in ruin theory,Insurance Math. Econom.,25, 63–84.

    Article  MATH  MathSciNet  Google Scholar 

  25. Lin, X. S. andWillmot, G. E., (2000). The moments of the time of ruin, the surplus before ruin, and the deficit at ruin,Insurance Math. Econom.,27, 19–44.

    Article  MATH  MathSciNet  Google Scholar 

  26. Michel, R., (1989). Representation of a time-discrete probability of eventual ruin,Insurance Math. Econom.,8, 149–152.

    Article  MATH  MathSciNet  Google Scholar 

  27. Neuts, M. F., (1975). Probability distributions of phase type, in:Liber Amicorum Prof. Emeritus H. Florin. University of Louvain, 173-206.

  28. Neuts, M. F., (1981).Matrix-geometric solutions in stochastic models: An algorithmic approach, Johns Hopkins University Press, Baltimore.

    MATH  Google Scholar 

  29. Pavlova, K. andWillmot, G. E., (2004). The discrete stationary renewal risk model and the Gerber-Shiu discounted penalty function,Insurance Math. Econom.,35, 267–277.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  31. Stanford, D. A. andStroiński, K. J., (1994). Recursive methods for computing finite-time ruin probabilities for phase-distributed claim sizes,Astin Bull.,24, 235–254.

    Article  Google Scholar 

  32. Willmot, G. E., (1993). Ruin probabilities in the compound binomial model,Insurance Math. Econom.,12, 133–142.

    Article  MATH  MathSciNet  Google Scholar 

  33. Willmot, G. E., (1999). A Laplace transform representation in a class of renewal queueing and risk processes,J. Appl. Probab.,36, 570–584.

    Article  MATH  MathSciNet  Google Scholar 

  34. Wu, X. and Li, S., (2008). On a discrete-time Sparre Anderson model with phase-type claims, Working paper 08-169, Department of Economics, 1–16.University of Melbourne, http://www.mercury.ecom.unimelb.edu.au/SITE/actwww/wps2008/No169.pdf

  35. Wu, X. and Li, S. (2009). On the Gerber-Shiu function in a discrete time renewal risk model with general inter-claim times,Scand. Actuar. J., to appear.

  36. Yang, H., Zhang, Z. andLan, C., (2009). Ruin problems in a discrete Markov risk model,Statistics and Probability Letters,79, 21–28.

    Article  MATH  MathSciNet  Google Scholar 

  37. Yuen, K. C. andGuo, J., (2001). Ruin probabilities for time-correlated claims in the compound binomial model,Insurance Math. Econom.,29, 47–57.

    Article  MATH  MathSciNet  Google Scholar 

  38. Yuen, K. C. andGuo, J., (2006). Some results on the compound binomial model,Scand. Actuar. J.,2006(3), 129–140.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre for Actuarial Studies, Department of Economics, The University of Melbourne, Australia

    Shuanming Li

  2. Department of Statistics and Actuarial Science, Simon Fraser University, Canada

    Yi Lu

  3. Department of Mathematics and Statistics, Concordia University, Canada

    José Garrido

Authors
  1. Shuanming Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yi Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José Garrido
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Li, S., Lu, Y. & Garrido, J. A review of discrete-time risk models. Rev. R. Acad. Cien. Serie A. Mat. 103, 321–337 (2009). https://doi.org/10.1007/BF03191910

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03191910

Keywords

Mathematics Subject Classifications

Search

Navigation