Transport plans with domain constraints

TL;DR

This paper studies constrained martingale optimal transport problems with bounded quadratic variation, providing existence criteria, extensions to multi-marginal cases, and duality results, with applications to volatility uncertainty and capacity constraints.

Contribution

It introduces new existence conditions for constrained martingale transport measures and extends duality theory to these problems, including multi-marginal constraints.

Findings

Characterization of measure existence under convex transport constraints

Extension of results to multi-marginal constraints

Derivation of Kantorovich duality and a geometric monotonicity principle

Abstract

This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain its Kantorovich duality. A corollary of this result is a monotonicity principle which gives a geometric way of identifying the optimizer.