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On Generalised Piterbarg Constants

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Abstract

We investigate generalised Piterbarg constants

$$\mathcal{P}_{\alpha, \delta}^{h}=\lim\limits_{T \rightarrow \infty} \mathbb{E}\left\{ \sup\limits_{t\in \delta \mathbb{Z} \cap [0,T]} e^{\sqrt{2}B_{\alpha}(t)-|t|^{\alpha}- h(t)}\right\} $$

determined in terms of a fractional Brownian motion B α with Hurst index α/2∈(0,1], the non-negative constant δ and a continuous function h. We show that these constants, similarly to generalised Pickands constants, appear naturally in the tail asymptotic behaviour of supremum of Gaussian processes. Further, we derive several bounds for \(\mathcal {P}_{\alpha , \delta }^{h}\) and in special cases explicit formulas are obtained.

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Authors and Affiliations

  1. Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015, Lausanne, Switzerland

    Long Bai, Enkelejd Hashorva & Li Luo

  2. Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50384, Wrocław, Poland

    Krzysztof Dȩbicki

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  1. Long Bai
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  2. Krzysztof Dȩbicki
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  3. Enkelejd Hashorva
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  4. Li Luo
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Correspondence to Enkelejd Hashorva.

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Bai, L., Dȩbicki, K., Hashorva, E. et al. On Generalised Piterbarg Constants. Methodol Comput Appl Probab 20, 137–164 (2018). https://doi.org/10.1007/s11009-016-9537-0

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  • DOI: https://doi.org/10.1007/s11009-016-9537-0

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