Weak decreasing stochastic order

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 Azéma-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.

On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales

“…Kellerer (1972) then showed that for ℝ-valued processes the martingale can in fact be chosen to be a Markov process. Recent contributions to the peacock literature include, next to the extensive work of Hirsch et al (2011), the articles by Bogso (2015Bogso ( , 2016, as well as Ewald and Yor (2015) who discuss applications in the economics of inequality. Further recent applications of peacocks involve Skorohod embeddings (see Källblad, Tan, & Touzi, 2017) and optimal transport under marginal constraints (see Beiglböck & Juillet, 2016).…”

On Peacocks and Lyrebirds: Australian Options, Brownian Bridges and the Average of Sub-Martingales

“…Kellerer (1972) then showed that for R-valued processes the martingale can in fact be chosen to be a Markov process. Recent contributions to the peacock literature include, next to the extensive work of Hirsch et al (2011), the articles by Bogso (2015Bogso ( , 2016, as well as Ewald and Yor (2015) who discuss applications in the economics of inequality. Further recent applications of peacocks involve Skorohod embeddings, see Källblad et al (2015) and optimal transport under marginal constraints, see Beiglböck and Juillet (2016).…”