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On Generalized Berman Constants

Author

Listed:
  • Chengxiu Ling

    (Xi’an Jiaotong-Liverpool University
    Laboratory for Intelligent Computing and Financial Technology)

  • Hong Zhang

    (Southwest University)

Abstract

Considering the important role in Gaussian related extreme value topics, we evaluate the Berman constants involved in the study of the sojourn time of Gaussian processes, given by B α h ( x , E ) = ∫ ℝ e z ℙ ∫ E I 2 B α ( t ) − | t | α − h ( t ) − z > 0 d t > x d z , x ∈ [ 0 , mes ( E ) ] , $$ \mathcal{B}_{\alpha}^{h}(x, E) = {\int}_{\mathbb{R}} e^{z} \mathbb{P} \left\{{{\int}_{E} \mathbb{I}\left( \sqrt2B_{\alpha}(t) - |t|^{\alpha} - h(t) - z>0 \right) \text{d} t \!>\! x}\right\} \text{d} z,\quad x\in[0, \text{mes}(E)], $$ where mes(E) is the Lebesgue measure of a compact set E ⊂ ℝ $E\subset \mathbb {R}$ , h is a continuous drift function, and Bα is a centered fractional Brownian motion (fBm) with Hurst index α/2 ∈ (0, 1]. This note specifies its explicit expression for α = 1 and α = 2 under certain conditions of drift functions. Explicit expressions of B 2 h ( x , E ) ${{\mathcal{B}}_{2}^{h}}(x, E)$ with typical drift functions are given and several bounds of B α h ( x , E ) ${\mathcal{B}}_{\alpha }^{h}(x, E)$ are established as well. Numerical studies are performed to illustrate the main results.

Suggested Citation

  • Chengxiu Ling & Hong Zhang, 2020. "On Generalized Berman Constants," Methodology and Computing in Applied Probability, Springer, vol. 22(3), pages 1125-1143, September.
  • Handle: RePEc:spr:metcap:v:22:y:2020:i:3:d:10.1007_s11009-019-09754-0
    DOI: 10.1007/s11009-019-09754-0
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    References listed on IDEAS

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    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. Debicki, Krzysztof, 2002. "Ruin probability for Gaussian integrated processes," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 151-174, March.
    3. Long Bai & Krzysztof Dȩbicki & Enkelejd Hashorva & Li Luo, 2018. "On Generalised Piterbarg Constants," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 137-164, March.
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