Existence of Transport Plans with Domain Constraints

13 Pages Posted: 1 May 2018  

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 Ω to be one of 𝕏N+1, C[0, 1], D[0, 1]: a product of Polish spaces, space of continuous functions from [0, 1] to a subset of ℝd, and space of RCLL (right-continuous with left limits) functions from [0, 1] to ℝd respectively. We consider the existence of a probability measure P on Ω such that P has the given marginals α and β and satisfies some other convex transport constraints, which is given by Γ. 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 α average of so-called G-expectation of bounded uniformly continuous and bounded functions with respect to the measures in Γ is less than their β 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 11, 2018). Available at SSRN: https://ssrn.com/abstract=3161652 or http://dx.doi.org/10.2139/ssrn.3161652

Erhan Bayraktar (Contact Author)

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Xin Zhang

University of Michigan at Ann Arbor - Department of Mathematics ( email )

2074 East Hall
530 Church Street
Ann Arbor, MI 48109-1043
United States

Zhou Zhou

University of Minnesota - Twin Cities ( email )

420 Delaware St. SE
Minneapolis, MN 55455
United States

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