On suprema of Lévy processes with light tails

Michael Braverman

doi:10.1016/j.spa.2009.12.004

On suprema of Lévy processes with light tails

Michael Braverman

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Let X (t), t ≥ 0, X (0) = 0, be a Lévy process with a spectral Lévy measure ρ. Assuming that ∫_{- 1}¹ | x | ρ (d x) < ∞ and the right tail of ρ is light, we show that in the presence of the Brownian component P (under(sup, 0 ≤ t ≤ 1) X (t) > u) ∼ P (X (1) > u) as u → ∞, while in the absence of a Brownian component these tails are not always comparable.

Original language	English
Pages (from-to)	541-573
Number of pages	33
Journal	Stochastic Processes and their Applications
Volume	120
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2009.12.004
State	Published - 1 Apr 2010

Keywords

Brownian motion
Poisson process
Supremum

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/j.spa.2009.12.004

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abstract = "Let X (t), t ≥ 0, X (0) = 0, be a L{\'e}vy process with a spectral L{\'e}vy measure ρ. Assuming that ∫- 11 | x | ρ (d x) < ∞ and the right tail of ρ is light, we show that in the presence of the Brownian component P (under(sup, 0 ≤ t ≤ 1) X (t) > u) ∼ P (X (1) > u) as u → ∞, while in the absence of a Brownian component these tails are not always comparable.",

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N2 - Let X (t), t ≥ 0, X (0) = 0, be a Lévy process with a spectral Lévy measure ρ. Assuming that ∫- 11 | x | ρ (d x) < ∞ and the right tail of ρ is light, we show that in the presence of the Brownian component P (under(sup, 0 ≤ t ≤ 1) X (t) > u) ∼ P (X (1) > u) as u → ∞, while in the absence of a Brownian component these tails are not always comparable.

AB - Let X (t), t ≥ 0, X (0) = 0, be a Lévy process with a spectral Lévy measure ρ. Assuming that ∫- 11 | x | ρ (d x) < ∞ and the right tail of ρ is light, we show that in the presence of the Brownian component P (under(sup, 0 ≤ t ≤ 1) X (t) > u) ∼ P (X (1) > u) as u → ∞, while in the absence of a Brownian component these tails are not always comparable.

KW - Brownian motion

KW - Poisson process

KW - Supremum

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U2 - 10.1016/j.spa.2009.12.004

DO - 10.1016/j.spa.2009.12.004

M3 - Article

AN - SCOPUS:76349084042

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SP - 541

EP - 573

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 4

ER -

On suprema of Lévy processes with light tails

Abstract

Keywords

ASJC Scopus subject areas

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