Weak decreasing stochastic order

TL;DR

This paper introduces the weak decreasing stochastic (WDS) order for processes with negative means, establishing its role in embedding such processes into Brownian motion and analyzing associated Markov properties.

Contribution

It defines the WDS order, proves its necessity and sufficiency for embedding in Brownian motion, and explores implications for Markovianity of quantile processes.

Findings

WDS order is necessary and sufficient for embedding in Brownian motion.

Cox-Hobson stopping times generate Markov processes for non-decreasing measures.

Quantile processes may not be Markovian despite the underlying process's properties.

Abstract

We introduce the notion of weak decreasing stochastic (WDS) ordering for real-valued processes with negative means, which, to our knowledge, has not been studied before. Thanks to Madan-Yor's argument, it follows that the WDS ordering is a necessary and sufficient condition for a process with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Az\'ema-Yor algorithm. Since the decreasing stochastic order is stronger than the WDS order, then, for every stochastically non-decreasing family of probability measures with densities, the Cox-Hobson stopping times provide an associated Markov process. The quantile process associated to a stochastically non-decreasing process is not necessarily Markovian.