Suprema of Lévy processes

Abstract

In this paper we study the supremum functional $M_t={\sup }_{0\le s\le t}{X}_s$, where $X_t$, $t\ge 0$, 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 $(0,\infty )$.

Published in at http://dx.doi.org/10.1214/11-AOP719 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Zooming-in on a Lévy process: Failure to observe threshold exceedance over a dense grid
• Hitting times of points and intervals for symmetric Lévy processes
• Geometrically Convergent Simulation of the Extrema of Lévy Processes
• First passage times of Lévy processes over a one-sided moving boundary
• Rogers functions and fluctuation theory
• Subordinated Markov processes: sharp estimates for heat kernels and Green functions
• First exit times from a bounded interval for Lévy processes
• The first passage time problem over a moving boundary for asymptotically stable Lévy processes
• On tails of exit times of multidimensional Lévy processes
• The unified form of Pollaczek--Khinchine formula for Lévy processes with matrix-exponential negative jumps