Weak decreasing stochastic order

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 family of integrable probability measures with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Azéma-Yor algorithm. The resulting process is a supermartingale and if, in addition, the measures have densities, this supermartingale is Markovian. Then the Cox-Hobson algorithm provides a special solution of Kellerer's theorem relying on the stronger hypothesis of WDS order.