The distribution of the supremum for spectrally asymmetric Lévy
processes
processes
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Published version
Author(s)
Michna, Z
Palmowski, Z
Pistorius, M
Type
Journal Article
Abstract
In this article we derive formulas for the probability $P(\sup_{t\leq T}
X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally
positive L\'evy process with infinite variation. The formulas are
generalizations of the well-known Tak\'acs formulas for stochastic processes
with non-negative and interchangeable increments. Moreover, we find the joint
distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally
negative L\'evy process.
X(t)>u)$ $T>0$ and $P(\sup_{t<\infty} X(t)>u)$ where $X$ is a spectrally
positive L\'evy process with infinite variation. The formulas are
generalizations of the well-known Tak\'acs formulas for stochastic processes
with non-negative and interchangeable increments. Moreover, we find the joint
distribution of $\inf_{t\leq T} Y(t)$ and $Y(T)$ where $Y$ is a spectrally
negative L\'evy process.
Date Issued
2015-03-13
Citation
Electronic Communications in Probability, 2015, 20
ISSN
1083-589X
Publisher
Institute of Mathematical Statistics (IMS)
Journal / Book Title
Electronic Communications in Probability
Volume
20
Copyright Statement
This work is licensed under a Creative Commons Attribution 3.0 License.
Identifier
http://arxiv.org/abs/1410.2554v2
Subjects
math.PR
math.PR
Publication Status
Published
Article Number
24