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Parisian ruin of self-similar Gaussian risk processes

  • Krzysztof Dębicki (a1), Enkelejd Hashorva (a2) and Lanpeng Ji (a2)
Abstract

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.

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Corresponding author
Postal address: Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland.
∗∗ Postal address: University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland.
∗∗∗ Email address: jilanpeng@126.com
References
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[1] Albin, J. M. P. and Choi, H. (2010). A new proof of an old result by Pickands. Electron. Commun. Prob. 15 339-345.
[2] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
[3] Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA.
[4] Chesney, M., Jeanblanc-Picqué, M. and Yor, M. (1997). Brownian excursions and Parisian barrier options. Adv. Appl. Prob. 29 165-184.
[5] Czarna, I. (2014). Parisian ruin probability with a lower ultimate bankrupt barrier. Scand. Actuarial J. 10.1080/03461238.2014.926288.
[6] Czarna, I. and Palmowski, Z. (2011). Ruin probability with Parisian delay for a spectrally negative Lévy risk process. J. Appl. Prob. 48 984-1002.
[7] 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.
[8] Czarna, I. and Palmowski, Z. (2014). Parisian quasi-stationary distributions for asymmetric Lévy processes. Preprint. Available at http://arxiv.org/abs/1404.3367.
[9] Czarna, I., Palmowski, Z. and Światek, P. (2014). Binomial discrete time ruin probability with Parisian delay. Preprint. Available at http://arxiv.org/abs/1403.7761.
[10] Dassios, A. and Wu, S. (2008). Parisian ruin with exponential claims. Preprint. Available at http://stats.lse.ac.uk/angelos/.
[11] Debicki, K. (2002). Ruin probability for Gaussian integrated processes. Stoch. Process. Appl. 98 151-174.
[12] Debicki, K. and Kisowski, P. (2008). A note on upper estimates for Pickands constants. Statist. Prob. Lett. 78 2046-2051.
[13] Debicki, K. and Kosiński, K. M. (2014). On the infimum attained by the reflected fractional Brownian motion. Extremes 17 431-446.
[14] Debicki, K., Hashorva, E. and Ji, L. (2015). Gaussian risk models with financial constraints. Scand. Actuarial J. 6, 469481.
[15] Debicki, K., Hashorva, E. and Ji, L. (2014). Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals. Extremes 17 411-429.
[16] Debicki, K., Michna, Z. and Rolski, T. (2003). Simulation of the asymptotic constant in some fluid models. Stoch. Models 19 407-423.
[17] Dieker, A. B. (2005). Extremes of Gaussian processes over an infinite horizon. Stoch. Process. Appl. 115 207-248.
[18] Dieker, A. B. and Yakir, B. (2014). On asymptotic constants in the theory of extremes for Gaussian processes. Bernoulli 20 1600-1619.
[19] Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton University Press.
[20] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. (New York) 33), Springer, Berlin.
[21] Griffin, P. S. (2013). Convolution equivalent Lévy processes and first passage times. Ann. Appl. Prob. 23 1506-1543.
[22] 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.
[23] Hashorva, E. and Ji, L. (2014). Approximation of passage times of γ-reflected processes with FBM input.break J. Appl. Prob. 51 713-726.
[24] Hashorva, E. and Ji, L. (2014). Extremes and first passage times of correlated fractional Brownian motions. Stoch. Models 30 272-299.
[25] Hashorva, E. and Ji, L. (2015). Piterbarg theorems for chi-processes with trend. Extremes 18 37-64.
[26] Hashorva, E., Ji, L. and Piterbarg, V. I. (2013). On the supremum of γ-reflected processes with fractional Brownian motion as input. Stoch. Process. Appl. 123 4111-4127.
[27] Hüsler, J. and Piterbarg, V. (1999). Extremes of a certain class of Gaussian processes. Stoch. Process. Appl. 83 257-271.
[28] 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.
[29] Hüsler, J. and Zhang, Y. (2008). On first and last ruin times of Gaussian processes. Statist. Prob. Lett. 78 1230-1235.
[30] Hüsler, J., Piterbarg, V. and Rumyantseva, E. (2011). Extremes of Gaussian processes with a smooth random variance. Stoch. Process. Appl. 121 2592-2605.
[31] 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.
[32] Landriault, D., Renaud, J.-F. and Zhou, X. (2014). An insurance risk model with Parisian implementation delays. Methodol. Comput. Appl. Prob. 16 583-607.
[33] Loeffen, R., Czarna, I. and Palmowski, Z. (2013). Parisian ruin probability for spectrally negative Lévy processes. Bernoulli 19 599-609.
[34] Mandjes, M. (2007). Large Deviations for Gaussian Queues. John Wiley, Chichester.
[35] Michna, Z. (1998). Self-similar processes in collective risk theory. J. Appl. Math. Stoch. Analysis 11 429-448.
[36] Palmowski, Z. and Światek, P. (2014). A note on first passage probabilities of a Lévy process reflected at a general barrier. Preprint. Available at http://arxiv.org/abs/1403.1025.
[37] Pickands, J. III (1969). Upcrossing probabilities for stationary Gaussian processes. Trans. Amer. Math. Soc. 145 51-73.
[38] 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.
[39] Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields (Trans. Math. Monogr. 148), American Mathematical Society, Providence, RI.
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  • EISSN: 1475-6072
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