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Extremes of Gaussian processes with a smooth random variance

Author

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  • Hösler, Jörg
  • Piterbarg, Vladimir
  • Rumyantseva, Ekaterina

Abstract

Let ξ(t) be a standard stationary Gaussian process with covariance function r(t), and η(t), another smooth random process. We consider the probabilities of exceedances of ξ(t)η(t) above a high level u occurring in an interval [0,T] with T>0. We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ(t)η(t), which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X(t) given a random element Y.

Suggested Citation

  • Hösler, Jörg & Piterbarg, Vladimir & Rumyantseva, Ekaterina, 2011. "Extremes of Gaussian processes with a smooth random variance," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2592-2605, November.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:11:p:2592-2605
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    References listed on IDEAS

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    1. Nielsen, Lars Tyge, 1999. "Pricing and Hedging of Derivative Securities," OUP Catalogue, Oxford University Press, number 9780198776192.
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    Cited by:

    1. Ji, Lanpeng & Peng, Xiaofan, 2023. "Extreme value theory for a sequence of suprema of a class of Gaussian processes with trend," Stochastic Processes and their Applications, Elsevier, vol. 158(C), pages 418-452.
    2. Goran Popivoda & Siniša Stamatović, 2024. "Sojourn Times of Gaussian Processes with Random Parameters," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2023-2053, September.
    3. Popivoda, Goran & Stamatović, Siniša, 2016. "Extremes of Gaussian fields with a smooth random variance," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 185-190.
    4. Tan, Zhongquan, 2013. "An almost sure limit theorem for the maxima of smooth stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2135-2141.

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