# Transport Plans with Domain Constraints

21 Pages Posted: 1 May 2018 Last revised: 24 Sep 2018

See all articles by Erhan Bayraktar

## 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 11, 2018

### Abstract

Let $\Omega$ be one of $\X^{N 1},C[0,1],D[0,1]$: product of Polish spaces, space of continuous functions from $[0,1]$ to $\mathbb{R}^d$, and space of RCLL (right-continuous with left limits) functions from $[0,1]$ to $\mathbb{R}^d$, respectively. We first consider the existence of a probability measure $P$ on $\Omega$ such that $P$ has the given marginals $\alpha$ and $\beta$ and its disintegration $P_x$ must be in some fixed $\Gamma(x) \subset \kP(\Omega)$, where $\kP(\Omega)$ is the set of probability measures on $\Omega$. The main application we have in mind is the martingale optimal transport problem 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 the so-called $G$-expectation of bounded continuous 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. Second, we consider the optimal transport problem with constraints and obtain the Kantorovich duality. A corollary of this result is a monotonicity principle which gives us a geometric way of identifying the optimizer.

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

Suggested Citation

Bayraktar, Erhan and Zhang, Xin and Zhou, Zhou, Transport Plans with Domain Constraints (April 11, 2018). Available at SSRN: https://ssrn.com/abstract=3161652 or http://dx.doi.org/10.2139/ssrn.3161652

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