From reflected L\'evy processes to stochastically monotone Markov processes via generalized inverses and supermodularity

TL;DR

This paper generalizes known monotonicity, convexity, and concavity properties from reflected Lévy processes to a broader class of stochastically monotone Markov processes using supermodularity and generalized inverses.

Contribution

It extends key monotonicity and convexity results to general stochastically monotone Markov processes, beyond reflected Lévy processes, via a unified framework.

Findings

Monotonicity results for supermodular functions of Markov processes.

Convexity and concavity properties in the transient and stationary cases.

Applicability to various common Markovian models.

Abstract

It was recently proven that the correlation function of the stationary version of a reflected L\'evy 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 nonnegative, 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 L\'evy 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 L\'evy 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.