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From reflected Lévy processes to stochastically monotone Markov processes via generalized inverses and supermodularity

Part of: Operations research and management science Markov processes Stochastic processes

Published online by Cambridge University Press:  19 September 2022

Offer Kella Open the ORCID record for Offer Kella [Opens in a new window]  and
Michel Mandjes
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Michel Mandjes*
Affiliation:
University of Amsterdam
*
*Postal address: Department of Statistics and Data Science, the Hebrew University of Jerusalem, Jerusalem 9190501, Israel. Email address: offer.kella@huji.ac.il
**Postal address: Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands. Email address: m.r.h.mandjes@uva.nl
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Abstract

It was recently proven that the correlation function of the stationary version of a reflected Lévy process is nonnegative, nonincreasing, and convex. In another branch of the literature it was established that the mean value of the reflected process starting from zero is nondecreasing and concave. In the present paper it is shown, by putting them in a common framework, that these results extend to substantially more general settings. Indeed, instead of reflected Lévy processes, we consider a class of more general stochastically monotone Markov processes. In this setup we show monotonicity results associated with a supermodular function of two coordinates of our Markov process, from which the above-mentioned monotonicity and convexity/concavity results directly follow, but now for the class of Markov processes considered rather than just reflected Lévy processes. In addition, various results for the transient case (when the Markov process is not in stationarity) are provided. The conditions imposed are natural, in that they are satisfied by various frequently used Markovian models, as illustrated by a series of examples.

Keywords

Stochastic storage processLévy-driven queuesSkorokhod problemmonotone and convex autocorrelationconcave mean

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 60G51: Processes with independent increments; L'evy processes 90B05: Inventory, storage, reservoirs

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 1 , March 2023 , pp. 68 - 84
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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