# Existence of Transport Plans with Domain Constraints

13 Pages Posted:

## Erhan Bayraktar

University of Michigan at Ann Arbor - Department of Mathematics

## Xin Zhang

University of Michigan at Ann Arbor - Department of Mathematics

## Zhou Zhou

University of Minnesota - Twin Cities

Date Written: April 12, 2018

### Abstract

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.

Keywords: Option Pricing, Strassen's Theorem, Kellerer's Theorem, Martingale optimal transport, domain constraints, bounded volatility/quadratic variation, $G$-expectations

Suggested Citation

Bayraktar, Erhan and Zhang, Xin and Zhou, Zhou, Existence of Transport Plans with Domain Constraints (April 12, 2018). Available at SSRN: https://ssrn.com/abstract=

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