Dual attainment for the martingale transport problem

Abstract

We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [5] established such existence for weak (quasi-sure) duality, [2] showed existence for the natural stronger (pointwise) duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when y → c(x, y) is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost c(x, y) is twice continuously differentiable in y. Further, this may not be improved as we give examples with c(x, ·) ∈ C 2−ε , ε > 0, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if c is Lipschitz.

Computational methods for martingale optimal transport problems

“…Given ε ≥ 0, let M ε (µ) ⊂ P(µ) be the subset containing all ε−approximating martingale measures. Then M ε (µ) is convex and closed with respect to the weak topology by (6), and thus compact. For a measurable function c : Ω N → R, the relaxed MOT problem is defined by…”

“…It remains to estimate P c R L (µ , ν ) − P c R L (µ, ν) . Recall that, in view of Remark 2.6 of [6],…”

A Benamou–Brenier formulation of martingale optimal transport

“…Is such an extremely weak notion to (PME) sufficient to obtain a reasonable theory of well-posedness? This alternative formulation of the dual problem is also very natural, as it takes the irreducible components into account (see [BJ16] for the definition and use of irreducible components for martingale transport, and [BNT17,BLO17] for an application to duality). This can be seen by writing Developing a good understanding of this phenomenon from the PDE point of view could be particularly interesting for the higher dimensional case, which is far more complicated since it is unclear which functional should replace the Newtonian potential, see [DMT17,OS17].…”

“…Related literature. The one-dimensional discrete-time martingale optimal transport problem is by now well understood due to the seminal work of [BJ16] for the geometric characterization of optimizers and [BNT17] for a complete duality theory, see also [BLO17]. This was extended to cover the discrete-time multi-marginal problem in [NST17].…”

A new class of costs for optimal transport planning

“…In [7], the authors propose a quasi-sure formulation for the dual problem for which they prove the existence of a maximiser, and also provide several examples and counter-examples. Still on the real line, [6] provides regularity hypotheses on the cost c which ensure the existence of point-wise (as opposed to quasi-sure) minimisers for the usual formulation of the martingale dual problem. The d-dimensional case for the cost c(x, y) = ± |x − y| is addressed in the remarkable paper [10], where the existence of a dual maximiser and the structure of the optimal martingale plans are described under the hypothesis that the measures μ and ν are in subharmonic order.…”