Conditional convex orders and measurable martingale couplings

Strassen’s classical martingale coupling theorem states that two random vectors are ordered in the convex (resp. increasing convex) stochastic order if and only if they admit a martingale (resp. submartingale) coupling. By analysing 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 random vectors conditioned on a random element taking values in a general measurable space. We provide an analogue of the conditional martingale coupling theorem in the language of probability kernels, and discuss how it 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.