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The supremum distribution of a Lévy process with no negative jumps

Published online by Cambridge University Press:  01 July 2016

J. Michael Harrison
J. Michael Harrison*
Affiliation:
Stanford University
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Abstract

Let be a process with stationary, independent increments and no negative jumps. Let be this same process modified by a reflecting barrier at zero (a storage process). Assuming that – and denote by ψ(s) the exponent function of X. A simple formula is derived for the Laplace transform of as a function of W(0). Using the fact that the distribution of M is the unique stationary distribution of the Markov process W, this yields an elementary proof that the Laplace transform of M is µs/ψ(s). If it follows that These surprisingly simple formulas were originally obtained by Zolotarev using analytical methods.

Keywords

STATIONARY INDEPENDENT INCREMENTSSTORAGE PROCESSESMARKOV PROCESSESREFLECTING BARRIERSLÉVY PROCESSES

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 2 , June 1977, pp. 417 - 422
Copyright
Copyright © Applied Probability Trust 1977 

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References

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