Existence of Transport Plans with Domain Constraints
13 Pages Posted: 1 May 2018
Date Written: April 11, 2018
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
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