Advertisement

Extremes

pp 1–23 | Cite as

The time of ultimate recovery in Gaussian risk model

  • Krzysztof Dȩbicki
  • Peng LiuEmail author
Article
  • 5 Downloads

Abstract

We analyze the distance \(\mathcal {R}_{T}(u)\) between the first and the last passage time of {X(t) − ct : t ∈ [0, T]} at level u in time horizon T ∈ (0, ], where X is a centered Gaussian process with stationary increments and \(c\in {\mathbb {R}}\), given that the first passage time occurred before T. Under some tractable assumptions on X, we find Δ(u) and G(x) such that
$$\lim\limits_{u\to\infty}\mathbb{P} \left( \mathcal{R}_{T}(u)>{\Delta}(u)x \right)=G(x), $$
for x ≥ 0. We distinguish two scenarios: T < and T = , that lead to qualitatively different asymptotics. The obtained results provide exact asymptotics of the ultimate recovery time after the ruin in Gaussian risk model.

Keywords

Gaussian risk process Exact asymptotics First ruin time Last ruin time Generalized Pickands-Piterbarg constant 

AMS 2000 Subject Classifications

Primary 60G15 Secondary 60G70 60K25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank Enkelejd Hashorva for discussions and comments that improved presentation of the results of this contribution. K. Dȩbicki was partially supported by NCN Grant No 2015/17/B/ST1/01102 (2016-2019) whereas P. Liu was supported by the Swiss National Science Foundation Grant 200021-175752/1.

References

  1. Bai, L., Dȩbicki, K., Hashorva, E., Luo, L.: On generalised Piterbarg constants. Comp. Meth. Appl. Prob. 20(1), 137–164 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chesney, M., Jeanblanc-Picqué, M., Yor, M.: Brownian excursions and Parisian barrier options. Adv. Appl. Probab. 29(1), 165–184 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chiu, S., Yin, C.: Passage times for a spectrally negative lévy process with applications to risk theory. Bernoulli 11(3), 511–522 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Dȩbicki, K., Hashorva, E., Ji, L.: Parisian ruin of self-similar Gaussian risk processes. J. Appl Probab. 52(3), 688–702 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dȩbicki, K., Hashorva, E., Ji, L.: Parisian ruin over a finite-time horizon. Sci. China Math. 59(3), 557–572 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Dȩbicki, K., Hashorva, E., Liu, P.: Uniform tail approximation of homogenous functionals of Gaussian fields. Adv. Appl. Probab. 49, 1037–1066 (2017)MathSciNetCrossRefGoogle Scholar
  7. Dȩbicki, K., Hashorva, E., Liu, P.: Extremes of γ-reflected Gaussian process with stationary increments. ESAIM Probab. Statist. 21, 495–535 (2017)CrossRefGoogle Scholar
  8. Dȩbicki, K., Kosiński, K.: On the infimum attained by the reflected fractional Brownian motion. Extremes 17(3), 431–446 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dȩbicki, K., Liu, P.: Extremes of stationary Gaussian storage models. Extremes 19(2), 273–302 (2016)MathSciNetCrossRefGoogle Scholar
  10. Dȩbicki, K.: Ruin probability for Gaussian integrated processes. Stochastic Process. Appl. 98(1), 151–174 (2002)MathSciNetCrossRefGoogle Scholar
  11. Dieker, A.B.: Extremes of Gaussian processes over an infinite horizon. Stochastic Process. Appl. 115(2), 207–248 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dieker, A.B., Mikosch, T.: Exact simulation of brown-Resnick random fields at a finite number of locations. Extremes 18, 301–314 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dieker, A.B., Yakir, B.: On asymptotic constants in the theory of Gaussian processes. Bernoulli 20(3), 1600–1619 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Frostig, E.: Upper bounds on the expected time to ruin and on the expected recovery time. Adv. Appl. Probab. 36(2), 377–397 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. Guérin, H., Renaud, J.-F.: On the distribution of cumulative Parisian ruin. Insurance Math. Econom. 73, 116–123 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. Harper, A.J.: Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab 23, 584–616 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Harper, A.J.: Pickands’ constant h α does not equal 1/γ(1/α), for small α. Bernoulli 23(1), 582–602 (2017)MathSciNetCrossRefGoogle Scholar
  18. Hashorva, E., Ji, L.: Approximation of passage times of γ-reflected processes with fBm input. J. Appl Probab. 51(3), 713–726 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Hüsler, J., Piterbarg, V.I.: On the ruin probability for physical fractional Brownian motion. Stochastic Process. Appl. 113(2), 315–332 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Hüsler, J., Piterbarg, V.I.: A limit theorem for the time of ruin in a Gaussian ruin problem. Stochastic Process. Appl. 118(11), 2014–2021 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Hüsler, J., Zhang, Y.: On first and last ruin times of Gaussian processes. Statist. Probab. Lett. 78(10), 1230–1235 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Iglehart, D.L.: Diffusion approximations in collective risk theory. J. Appl. Probability 6, 285–292 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Li, S.: The time of recovery and the maximum severity of ruin in a Sparre Andersen model. N. Am. Actuar. J. 12(4), 413–427 (2008)MathSciNetCrossRefGoogle Scholar
  24. Liu, P., Hashorva, E., Ji, L.: On the γ-reflected processes with fBm input. Lithuanian Math J. 55(3), 402–412 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Loeffen, R., Czarna, I., Palmowski, Z.: Parisian ruin probability for spectrally negative lévy processes. Bernoulli 19(2), 599–609 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  26. Michna, Z.: Self-similar processes in collective risk theory. J. Appl. Math. Stochastic Anal. 11(4), 429–448 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. Piterbarg, V.I.: Asymptotic methods in the theory of Gaussian processes and fields, volume 148 of translations of mathematical monographs. American Mathematical Society, Providence, RI (1996)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial Science, Faculty of Business and EconomicsUniversity of LausanneLausanneSwitzerland

Personalised recommendations