Complete duality for martingale optimal transport on the line

Abstract

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

“…For this quantity to be non-zero the following have to happen prior to t: first B (1) has to hit a without reaching b, then it has to come back to x = 0 and continue to y without ever reaching c. This happens at time ρ (3) + H (4) y and from then onwards the local time L y, (1) is counted before time t ∧ H (1) c,b and we see that it simply corresponds to L 0, (5) . With a similar reasoning for L y, (2) , we see that our construction gives us the desired coupling:…”

“…The connection with optimal stopping is examined in Sect. 5. Given this preparation, we report the proof of the main result in Sect.…”

“…Consider three independent Brownian motions B (3) , B (4) , B (5) (4) y )∧ρ (3) + B (5) t−t∧(ρ (3) +H (4) y ) (6.1) and observe these are standard Brownian motions. This construction is depicted in Fig.…”

“…Note that this might be so even if ψ(x)μ i (dx) = ∞ for each 0 ≤ i ≤ n. More generally, thanks to the convex ordering of measures, one can define the integral g(x)(μ j − μ i )(dx) for a convex g and 0 ≤ i ≤ j ≤ n. This is done by considering g k g which are convex, equal to g on a compact set and affine on the complement. Further, if h = h − g + g with (h − g) and g convex with finite integrals against (μ j − μ i ) then the integral h(x)(μ j − μ i )(dx) is also well defined and finite, see Beiglböck et al [5] for details. We shall use this fact below repeatedly together with ψ(x)(μ j − μ i )(dx) < ∞ which follows from (A.10).…”

Optimal Skorokhod Embedding Under Finitely Many Marginal Constraints

“…More recently, by relaxing the dual formulation of the one dimensional discrete-time martingale transport, the existence of the dual optimizer in "weak" sense is obtained by Beiglböck, Nutz & Touzi [5]. [34] for a detailed review on these constructions.…”