Journal of Applied Probability

Parisian ruin of self-similar Gaussian risk processes

Krzysztof Dębicki, Enkelejd Hashorva, and Lanpeng Ji

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In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.

Article information

J. Appl. Probab. Volume 52, Number 3 (2015), 688-702.

First available in Project Euclid: 22 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes

Parisian ruin time Parisian ruin probability self-similar Gaussian process fractional Brownian motion normal approximation generalized Pickands' constant


Dębicki, Krzysztof; Hashorva, Enkelejd; Ji, Lanpeng. Parisian ruin of self-similar Gaussian risk processes. J. Appl. Probab. 52 (2015), no. 3, 688--702. doi:10.1239/jap/1445543840.

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  • Albin, J. M. P. and Choi, H. (2010). A new proof of an old result by Pickands. Electron. Commun. Prob. 15, 339–345.
  • Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
  • Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
  • Chesney, M., Jeanblanc-Picqué, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29, 165–184.
  • Czarna, I. (2014). Parisian ruin probability with a lower ultimate bankrupt barrier. Scand. Actuarial J. 10.1080/03461238.2014.926288.
  • Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Prob. 48, 984–1002.
  • Czarna, I. and Palmowski, Z. (2014). Dividend problem with Parisian delay for a spectrally negative Lévy risk process. J. Optimization Theory Appl. 161, 239–256.
  • Czarna, I. and Palmowski, Z. (2014). Parisian quasi-stationary distributions for asymmetric Lévy processes. Preprint. Available at
  • Czarna, I., Palmowski, Z. and Świątek, P. (2014). Binomial discrete time ruin probability with Parisian delay. Preprint. Available at
  • Dassios, A. and Wu, S. (2008). Parisian ruin with exponential claims. Preprint. Available at
  • Dębicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98, 151–174.
  • Dębicki, K. and Kisowski, P. (2008). A note on upper estimates for Pickands constants. Statist. Prob. Lett. 78, 2046–2051.
  • Dębicki, K. and Kosiński, K. M. (2014). On the infimum attained by the reflected fractional Brownian motion. Extremes 17, 431–446.
  • Dębicki, K., Hashorva, E. and Ji, L. (2015). Gaussian risk models with financial constraints. Scand. Actuarial J. \textbf 6, 469–481.
  • Dębicki, K., Hashorva, E. and Ji, L. (2014). Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals. Extremes 17, 411–429.
  • Dębicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19, 407–423.
  • Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115, 207–248.
  • Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20, 1600–1619.
  • Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. (New York) 33). Springer, Berlin.
  • Griffin, P. S. (2013). Convolution equivalent Lévy processes and first passage times. Ann. Appl. Prob. 23, 1506–1543.
  • Griffin, P. S. and Maller, R. A. (2012). Path decomposition of ruinous behavior for a general Lévy insurance risk process. Ann. Appl. Prob. 22, 1411–1449.
  • Hashorva, E. and Ji, L. (2014). Approximation of passage times of $\gamma$-reflected processes with FBM input. J. Appl. Prob. 51, 713–726.
  • Hashorva, E. and Ji, L. (2014). Extremes and first passage times of correlated fractional Brownian motions. Stoch. Models 30, 272–299.
  • Hashorva, E. and Ji, L. (2015). Piterbarg theorems for chi-processes with trend. Extremes 18, 37–64.
  • Hashorva, E., Ji, L. and Piterbarg, V. I. (2013). On the supremum of $\gamma$-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123, 4111–4127.
  • Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83, 257–271.
  • Hüsler, J. and Piterbarg, V. (2008). A limit theorem for the time of ruin in a Gaussian ruin problem. Stoch. Process. Appl. 118, 2014–2021.
  • Hüsler, J. and Zhang, Y. (2008). On first and last ruin times of Gaussian processes. Statist. Prob. Lett. 78, 1230–1235.
  • Hüsler, J., Piterbarg, V. and Rumyantseva, E. (2011). Extremes of Gaussian processes with a smooth random variance. Stoch. Process. Appl. 121, 2592–2605.
  • Klüppelberg, C. and Kühn, C. (2004). Fractional Brownian motion as a weak limit of Poisson shot noise processes–-with applications to finance. Stoch. Process. Appl. 113, 333–351.
  • Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Prob. 16, 583–607.
  • Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19, 599–609.
  • Mandjes, M. (2007). Large Deviations for Gaussian Queues. John Wiley, Chichester.
  • Michna, Z. (1998). Self-similar processes in collective risk theory. J. Appl. Math. Stoch. Analysis 11, 429–448.
  • Palmowski, Z. and Świątek, P. (2014). A note on first passage probabilities of a Lévy process reflected at a general barrier. Preprint. Available at
  • Pickands, J., III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145, 51–73.
  • Piterbarg, V. I. (1972). On the paper by J. Pickands “Upcrossing probabilities for stationary Gaussian processes”. Vestnik Moskov. Univ. Ser. I Mat. Meh. 27, 25–30.
  • Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Trans. Math. Monogr. 148). American Mathematical Society, Providence, RI.