SUPREMA OF LÉVY PROCESSES

SUPREMA OF LÉVY PROCESSES

In this paper we study the supremum functional M t = sup 0≤s≤t X s , where X t , t ≥ 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 th...

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Veröffentlicht in:The Annals of probability 2013-05, Vol.41 (3), p.2047-2065
Hauptverfasser: Kwaśnicki, Mateusz, Małecki, Jacek, Ryznar, Michał
Format: Artikel
Sprache:eng
Schlagworte:
60E10
60G51
60J75
Brownian motion
Dimensional analysis
Estimating techniques
first passage time
Fluctuation theorem
fluctuation theory
Infinity
Laplace transformation
Laplace transforms
Lévy process
Mathematical functions
Mathematical theorems
Poisson process
Potential theory
Probability distribution
Statistics
Studies
supremum functional
Symmetry
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Beschreibung
Zusammenfassung:In this paper we study the supremum functional M t = sup 0≤s≤t X s , where X t , t ≥ 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, ∞).
ISSN:0091-1798
2168-894X
DOI:10.1214/11-AOP719