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Concavity and reflected Lévy processes

Part of: Stochastic analysis Stochastic processes Markov processes Real functions

Published online by Cambridge University Press:  14 July 2016

Offer Kella
Offer Kella*
Affiliation:
Yale University
*
∗ Postal address: Department of Operations Research, Yale University, 84 Trumbull Street, New Haven, CT 06511, USA.
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Abstract

Simple necessary and sufficient conditions for a function to be concave in terms of its shifted Laplace transform are given. As an application of this result, we show that the expected local time at zero of a reflected Lévy process with no negative jumps, starting from the origin, is a concave function of the time variable. A special case is the expected cumulative idle time in an M/G/1 queue. An immediate corollary is the concavity of the expected value of the reflected Lévy process itself. A special case is the virtual waiting time in an M/G/1 queue.

Keywords

LOCAL TIMEMARTINGALESTOCHASTIC INTEGRATIONM/G/1 QUEUEIDLE TIME

MSC classification

Secondary: 26A51: Convexity, generalizations 60H05: Stochastic integrals 60G44: Martingales with continuous parameter 60J55: Local time and additive functionals
Type
Short Communications
Information
Journal of Applied Probability , Volume 29 , Issue 1 , March 1992 , pp. 209 - 215
Copyright
Copyright © Applied Probability Trust 1992 

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