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arXiv · 1103.0935

Suprema of Lévy processes

Mateusz Kwaśnicki·Jacek Małecki·Michał Ryznar
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Abstract

In this paper we study the supremum functional $M_t=\sup_{0\le s\le t}X_s$, where $X_t$, $t\ge0$, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of $M_t$. In the symmetric case we find an integral representation of the Laplace transform of the distribution of $M_t$ if the Lévy-Khintchin exponent of the process increases on $(0,\infty)$.

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Mateusz Kwaśnicki, Jacek Małecki, Michał Ryznar. 2013-07-08. Suprema of Lévy processes. https://doi.org/10.1214/11-aop719

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