Existence of transport plans with domain constraints. (arXiv:1804.04283v1 [math.PR])

Let $\Omega$ to be one of $\X^{N+1},C[0,1],D[0,1]$: a product of Polish spaces, space of continuous functions from $[0,1]$ to a subset of $\mathbb{R}^d$, and space of RCLL (right-continuous with left limits) functions from $[0,1]$ to $\mathbb{R}^d$ respectively. We consider the existence of a probability measure $P$ on $\Omega$ such that $P$ has the given marginals $\alpha$ and $\beta$ and satisfies some other convex transport constraints, which is given by $\Gamma$. The main application we have in mind is the martingale optimal transport problem with when the martingales are assumed to have bounded volatility/quadratic variation. We show that such probability measure exists if and only if the $\alpha$ average of so-called $G$-expectation of bounded uniformly continuous and bounded functions with respect to the measures in $\Gamma$ is less than their $\beta$ average. As a byproduct, we get a necessary and sufficient condition for the Skorokhod embedding for bounded stopping times. 查看全文>>