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

Author

Listed:
  • Riccardo Gatto

    (University of Bern)

  • Benjamin Baumgartner

    (University of Bern)

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.

Suggested Citation

  • Riccardo Gatto & Benjamin Baumgartner, 2016. "Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 217-235, March.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:1:d:10.1007_s11009-014-9412-9
    DOI: 10.1007/s11009-014-9412-9
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    References listed on IDEAS

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    1. Riccardo Gatto & Benjamin Baumgartner, 2014. "Value at Ruin and Tail Value at Ruin of the Compound Poisson Process with Diffusion and Efficient Computational Methods," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 561-582, September.
    2. Feng, Runhuan, 2011. "An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 304-313, March.
    3. Dufresne, François & Gerber, Hans U., 1989. "Three Methods to Calculate the Probability of Ruin," ASTIN Bulletin, Cambridge University Press, vol. 19(1), pages 71-90, April.
    4. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    5. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "A generalized defective renewal equation for the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 51-66, February.
    6. Gerber, Hans U. & Landry, Bruno, 1998. "On the discounted penalty at ruin in a jump-diffusion and the perpetual put option," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 263-276, July.
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    Cited by:

    1. Youngjun Choe & Henry Lam & Eunshin Byon, 2018. "Uncertainty Quantification of Stochastic Simulation for Black-box Computer Experiments," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1155-1172, December.
    2. He, Yue & Kawai, Reiichiro, 2022. "Moment and polynomial bounds for ruin-related quantities in risk theory," European Journal of Operational Research, Elsevier, vol. 302(3), pages 1255-1271.

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