Skip to main content
SpringerLink
Log in

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

  • Published:
Methodology and Computing in Applied Probability Aims and scope Submit manuscript

Abstract

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.

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

  • Asmussen S (1985) Conjugate processes and the simulation of ruin problems. Stoch Process Appl 20(2):213–229

    Article  MathSciNet  MATH  Google Scholar 

  • Asmussen S, Albrecher H (2010) Ruin Probabilities, Vol. 14 of Advanced Series on Statistical Science and Applied Probability. World Scientific, Singapore

    MATH  Google Scholar 

  • Asmussen S, Glynn PW (2007) Stochastic Simulation: Algorithms and Analysis, number 57 in Stochastic Modelling and Applied Probability. Springer, New York

    MATH  Google Scholar 

  • Asmussen S, Rolski T (1991) Computational methods in risk theory: a matrix-algorithmic approach. Insur Math Econ 10(4):259–274

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE (1986) Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73(2):307–322

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE (1990) Approximate interval probabilities. J R Stat Soc: Series B 52(3):485–496

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE (1990b) A note on the standardized signed log likelihood ratio. Scand J Stat 17(2):157–160

    MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE (1991) Modified signed log likelihood ratio. Biometrika 78(3):557–563

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE, Schmidli H (1995) Saddlepoint approximation for the probability of ruin in finite time. Scand Actuar J 1995(2):169–186

    Article  MathSciNet  MATH  Google Scholar 

  • Biffis E, Morales M (2010) On a generalization of the Gerber-Shiu function to path dependent penalties. Insur: Math Econ 46(1):92–97

    MathSciNet  MATH  Google Scholar 

  • Daniels HE (1954) Saddlepoint approximations in statistics. Annals Math Stat 25(4):631–650

    Article  MathSciNet  MATH  Google Scholar 

  • Dietrich C, Newsam G (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix. SIAM J Sci Comput 18(4):1088–1107

    Article  MathSciNet  MATH  Google Scholar 

  • Dufresne F, GerberH-U (1989) Three methods to calculate the probability of ruin. ASTIN Bull 19(1):71–90

    Article  Google Scholar 

  • Dufresne F, Gerber H-U (1991) Risk theory for the compound Poisson process that is perturbed by diffusion. Insur: Math Econ 10(1):51–59

    MathSciNet  MATH  Google Scholar 

  • Feng R (2011) An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models. Insur: Math Econ 48(2):304–313

    MathSciNet  MATH  Google Scholar 

  • Feng R, Shimizu Y (2013) On a generalization from ruin to default in a Lévy insurance risk model. Methodol Comput Appl Probab 15(4):773–802

    Article  MathSciNet  MATH  Google Scholar 

  • Field C, Ronchetti E (1990) Small Sample Asymptotics, Vol. 13 of IMS Lecture Notes–Monograph Series, Institute of Mathematical Statistics, Hayward, CA

  • Gatto R, Baumgartner B (2014) Value at ruin and tail value at ruin of the compound Poisson process with diffusion and efficient computational methods. Methodol Comput ApplProbab online first 1–22

    MathSciNet  MATH  Google Scholar 

  • Gatto R, Jammalamadaka SR (1999) A conditional saddlepoint approximation for testing problems. J Am Stat Assoc 94(446):533–541

    Article  MathSciNet  MATH  Google Scholar 

  • Gatto R, Mosimann M (2012) Four approaches to compute the probability of ruin in the compound Poisson risk process with diffusion. Math Comput Model 55(3–4):1169–1185

    Article  MathSciNet  MATH  Google Scholar 

  • Gerber H.-U. (1970) An extension of the renewal equation and its application in the collective theory of risk. Skandinavisk Aktuarietidskrift 53(1):205–210

    MathSciNet  MATH  Google Scholar 

  • Gerber H.-U, Landry B (1998) On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insur: Math Econ 22(3):263–276

    MathSciNet  MATH  Google Scholar 

  • Gerber H.-U, Shiu E (1998) ‘On the time value of ruin’. North Am Actuar J 48(2):263–276

    MathSciNet  Google Scholar 

  • Jensen JL (1992) The modified signed likelihood statistic and saddlepoint approximations. Biometrika 79(4):693–703

    Article  MathSciNet  MATH  Google Scholar 

  • Jensen JL (1995) Saddlepoint Approximations, number 16 in Oxford Statistical Science Series. Oxford University Press, Oxford

    Google Scholar 

  • Li S, Garrido J (2005) The Gerber-Shiu function in a sparre andersen risk process pertrurbed by diffusion. Scand Actuar J 3:161–186

    Article  MathSciNet  MATH  Google Scholar 

  • Lugannani R, Rice S (1980) Saddle point approximation for the distribution of the sum of independent random variables. Adv Appl Prob 12(2):475–490

    Article  MathSciNet  MATH  Google Scholar 

  • Rolski T, Schmidli H, Schmidt V, Teugels JL (1999) Stochastic Processes for Insurance and Finance, number 505 in Wiley Series in Probability and Statistics. Wiley, New York

    Book  MATH  Google Scholar 

  • Schmidli H (1999) Perturbed risk processes: A review. Theory Stoch Process 5:145–165

    MathSciNet  MATH  Google Scholar 

  • Shimizu Y (2012) Nonparametric estimation of the Gerber-Shiu function for the Wiener-Poisson risk model. Scand Actuar J 1:56–69

    Article  MATH  Google Scholar 

  • Siegmund D (1976) Importance sampling in the Monte Carlo study of sequential tests. Annals Stat 4(4):673–684

    Article  MathSciNet  MATH  Google Scholar 

  • Skovgaard IM (1987) ‘Saddlepoint expansions for conditional distributions’. J Appl Prob 24(4):875–887

    Article  MathSciNet  MATH  Google Scholar 

  • Stanford DA, Yu K, Ren J (2011) Erlangian approximation to finite time ruin probabilities in perturbed risk models. Scand Actuar J 2011(1):38–58

    Article  MathSciNet  MATH  Google Scholar 

  • Tsai CC-L, Willmot GE (2002) A generalized defective renewal equation for the surplus process perturbed by diffusion. J R Stat Soc: Ser B 52(3):485–496

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, 3012, Bern, Switzerland

    Riccardo Gatto & Benjamin Baumgartner

Authors
  1. Riccardo Gatto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benjamin Baumgartner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Riccardo Gatto.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gatto, R., Baumgartner, B. Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion. Methodol Comput Appl Probab 18, 217–235 (2016). https://doi.org/10.1007/s11009-014-9412-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-014-9412-9

Keywords

AMS 2000 Subject Classification

Search

Navigation