Conditional convex orders and measurable martingale couplings

Understand

Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling.

• By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space.
• We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be applied in the analysis of pseudo-marginal Markov chain Monte Carlo algorithms.
• We also illustrate how our results imply the existence of a measurable minimiser in the context of martingale optimal transport.