On extrema of stable processes

Abstract

We study the Wiener--Hopf factorization and the distribution of extrema for general stable processes. By connecting the Wiener--Hopf factors with a certain elliptic-like function we are able to obtain many explicit and general results, such as infinite series representations and asymptotic expansions for the density of supremum, explicit expressions for the Wiener--Hopf factors and the Mellin transform of the supremum, quasi-periodicity and functional identities for these functions, finite product representations in some special cases and identities in distribution satisfied by the supremum functional.

Published in at http://dx.doi.org/10.1214/10-AOP577 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• The law of the supremum of a stable Lévy process with no negative jumps
• The asymptotic behavior of densities related to the supremum of a stable process
• On Wiener-Hopf factors for stable processes

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• Exact Simulation of the Extrema of Stable Processes
• Measurement-induced Lévy flights of quantum information
• On the Barnes double gamma function
• Joint density of a stable process and its supremum: regularity and upper bounds
• The first passage time of a stable process conditioned to not overshoot