Occupation time distributions for Lévy bridges and excursions

Let X be a one-dimensional Lévy process. It is shown that under the bridge law for X starting from 0 and ending at 0 at time t, the amount of time X spends positive has a uniform distribution on [0, t]. When 0 is a regular point, this uniform distribution result leads to an explicit expression for t...

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Bibliographic Details
Published inStochastic processes and their applications Vol. 58; no. 1; pp. 73 - 89
Main Authors Fitzsimmons, P.J., Getoor, R.K.
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier B.V 01.07.1995
Elsevier Science
Elsevier
SeriesStochastic Processes and their Applications
Subjects
Exact sciences and technology
Excursion
Lévy bridge
Lévy process
Lévy process Lévy bridge Excursion Occupation time Uniform distribution
Mathematics
Occupation time
Probability and statistics
Probability theory and stochastic processes
Sciences and techniques of general use
Stochastic processes
Uniform distribution
Lévy bridge
Excursion
Uniform distribution
Occupation time
Lévy process
Laplace transforms
Stable process
Joint distribution
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Summary:Let X be a one-dimensional Lévy process. It is shown that under the bridge law for X starting from 0 and ending at 0 at time t, the amount of time X spends positive has a uniform distribution on [0, t]. When 0 is a regular point, this uniform distribution result leads to an explicit expression for the Laplace transform of the joint distribution of the pair (R, AR), where R is the length of an excursion of X from 0, and AR is the total time X spends positive during the excursion. More concrete expressions are obtained for stable processes by specialization. In particular, a formula determining the distribution of ARR is given in the stable case.
ISSN:0304-4149
1879-209X
DOI:10.1016/0304-4149(95)00013-W