Monotone bivariate Markov kernels with specified marginals

Abstract:

Given two Markov kernels $k$ and $k’$ on an ordered Polish space, such that $k$ is stochastically dominated by $k’$, we establish the existence of: (i) a monotone bivariate Markov kernel whose marginals are $k$ and $k’$ and (ii) an upward coupler from $k$ to $k’$. This extends the results of Strassen, Kamae, Krengel and O’Brien to Markov kernels. Two examples are also given. The first is a simple illustration of our original motivation for this work, while the second demonstrates the optimality of our main result. The key technique is a combination of the standard probability/charge approach and the use of measurable selections of multivalued measurable maps.