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Ruin probability for Gaussian integrated processes

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  • Debicki, Krzysztof

Abstract

Pickands constants play an important role in the exact asymptotic of extreme values for Gaussian stochastic processes. By the generalized Pickands constant we mean the limitwhere and [eta](t) is a centered Gaussian process with stationary increments and variance function [sigma][eta]2(t). Under some mild conditions on [sigma][eta]2(t) we prove that is well defined and we give a comparison criterion for the generalized Pickands constants. Moreover we prove a theorem that extends result of Pickands for certain stationary Gaussian processes. As an application we obtain the exact asymptotic behavior of as u-->[infinity], where and Z(s) is a stationary centered Gaussian process with covariance function R(t) fulfilling some integrability conditions.

Suggested Citation

  • Debicki, Krzysztof, 2002. "Ruin probability for Gaussian integrated processes," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 151-174, March.
    • Handle: RePEc:eee:spapps:v:98:y:2002:i:1:p:151-174
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    References listed on IDEAS

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    1. Hüsler, J. & Piterbarg, V., 1999. "Extremes of a certain class of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 257-271, October.
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    Cited by:

    1. Dieker, A.B., 2005. "Extremes of Gaussian processes over an infinite horizon," Stochastic Processes and their Applications, Elsevier, vol. 115(2), pages 207-248, February.
    2. Dȩbicki, Krzysztof & Hashorva, Enkelejd & Ji, Lanpeng & Tabiś, Kamil, 2014. "On the probability of conjunctions of stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 141-148.
    3. Zhongquan Tan & Enkelejd Hashorva, 2014. "On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes," Methodology and Computing in Applied Probability, Springer, vol. 16(1), pages 169-185, March.
    4. Dëbicki, Krzysztof & Kisowski, Pawel, 2008. "A note on upper estimates for Pickands constants," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2046-2051, October.
    5. Hüsler, Jürg & Zhang, Yueming, 2008. "On first and last ruin times of Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 78(10), pages 1230-1235, August.
    6. Tan, Zhongquan & Hashorva, Enkelejd, 2013. "Exact asymptotics and limit theorems for supremum of stationary χ-processes over a random interval," Stochastic Processes and their Applications, Elsevier, vol. 123(8), pages 2983-2998.
    7. Chengxiu Ling & Hong Zhang, 2020. "On Generalized Berman Constants," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1125-1143, September.

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