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Extremes of threshold-dependent Gaussian processes

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Abstract

In this paper, we are concerned with the asymptotic behavior, as \(u\rightarrow\infty\), of \(\text{P}\left\{ {{{\sup }_{t \in [0,T]}}{X_u}(t) > u} \right\}\), where \(X_u(t),t\in[0,T],u>0\) is a family of centered Gaussian processes with continuous trajectories. A key application of our findings concerns \(\text{P}\left\{ {{{\sup }_{t \in [0,T]}}(X(t) + g(t)) > u} \right\}\), as \(u\rightarrow\infty\), for X a centered Gaussian process and g some measurable trend function. Further applications include the approximation of both the ruin time and the ruin probability of the Brownian motion risk model with constant force of interest.

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Acknowledgements

This work was supported by Swiss National Science Foundation (Grant No. 200021-166274) and the National Science Centre (Poland) (Grant No. 2015/17/B/ST1/01102) (2016–2019). The authors are thankful to the referees for several suggestions which have significantly improved their manuscript.

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Authors and Affiliations

  1. Department of Actuarial Science, University of Lausanne, Lausanne, 1015, Switzerland

    Long Bai & Enkelejd Hashorva

  2. Mathematical Institute, University of Wrocław, Wrocław, 50-384, Poland

    Krzysztof Dȩbicki

  3. School of Mathematics, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK

    Lanpeng Ji

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  1. Long Bai
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Correspondence to Krzysztof Dȩbicki.

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Bai, L., Dȩbicki, K., Hashorva, E. et al. Extremes of threshold-dependent Gaussian processes. Sci. China Math. 61, 1971–2002 (2018). https://doi.org/10.1007/s11425-017-9225-7

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