  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.
