Optimal transport and Skorokhod embedding

Abstract

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to aThe first author was supported by the FWF-Grants P26736 and Y00782, the third author by the CRC 1060.

Optimal Skorokhod Embedding Under Finitely Many Marginal Constraints

“…The above dual formulation (2.6) has been initially provided and proved by [1] in the one marginal case (n = 1), under the condition that (ω, θ) → Φ(ω θ∧· , θ) is bounded from above and upper semicontinuous. When n = 1, our duality results hold under more general conditions: Φ is non-anticipative, bounded from above, and θ → Φ(ω θ∧· , θ) is u.s.c.…”

“…The dual problem in [1,2] uses a pathwise formulation. Moreover, instead of the stochastic integral (H · B) in our case, they use martingales which are continuous in (t, ω) in the dual formulation.…”

“…Remark 2.7. A characterization of the optimizers P * has been provided in [1], called monotonicity principle. An alternative proof of this result is given in our accompanying paper [28].…”

“…Optimal SEP As in Beiglböck, Cox & Huesmann [1], we shall consider the problem in a weak setting, i.e. the stopping times may be identified by probability measures on the enlarged space Ω.…”

On the Monotonicity Principle of Optimal Skorokhod Embedding Problem

“…We provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established in Beiglböck, Cox & Huesmann [2]. Our proof is based on the adaptation of the Monge-Kantorovich duality in our context, a delicate application of the optional cross-section theorem, and a clever conditioning argument introduced in [2]. …”