Fine properties of the optimal Skorokhod embedding problem

Abstract

We study the problem of stopping a Brownian motion at a given distribution \nu while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set \mathcal{T}(\nu) of stopping times embedding \nu is weakly dense in the set \mathcal{R}(\nu) of randomized embeddings. In particular, the optimal Skorokhod embedding problem over \mathcal{T}(\nu) has the same value as the relaxed one over \mathcal{R}(\nu) when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem that exploits a novel compactness property and yields existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglböck, Cox and Huesmann [Optimal transport and Skorokhod embedding, Invent. Math. 208, 327–400 (2017)]. We show that these results can be applied to characterize the geometry of optimal embeddings through a variational condition.

A non-linear monotonicity principle and applications to Schrödinger type problems

“…Recent developments and aims of this article. More recently, variants of this 'monotonicity priniciple' have been applied in transport problems for finitely or infinitely many marginals [38,19,27,8,45], the martingale version of the optimal transport problem [9,36,11], stochastic portfolio theory [37], the Skorokhod embedding problem [5,28], the distribution constrained optimal stopping problem [6,10] and the weak transport problem [26,3,4].…”

Stability of Entropic Optimal Transport and Schrödinger Bridges

“…See also [29] which studies a related asymptotic regime for a different regularization of optimal transport. Related to the present work at least in spirit, there are several areas where analogues of c-cyclical monotonicity have recently lead to breakthroughs, including martingale optimal transport [7], optimal Skorokhod embeddings [5,8] and weak transport [4].…”

A non‐linear monotonicity principle and applications to Schrödinger‐type problems

“…More recently, variants of this 'monotonicity principle' have been applied in transport problems for finitely or infinitely many marginals [8,19,27,38,45], the martingale version of the optimal transport problem [9,11,36], stochastic portfolio theory [37], the Skorokhod embedding problem [5,28], the distribution constrained optimal stopping problem [6,10] and the weak transport problem [3,4,26]. What all these articles have in common is that the original idea is applied to other infinitedimensional linear optimization problems.…”