Optimal transport and Skorokhod embedding

By Mathias Beiglbock, Alexander M. G. Cox, Martin Huesmann. Inventiones Mathematicae, 2013

162 citations recorded by the source corpus.

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 a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/dd362b18dfca0dce2d286b02b0c0b8fe26d98a98

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Citations in the corpus

Stability of entropic optimal transport and Schrödinger bridges (2021, background): “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].”
The Wasserstein space of stochastic processes (2021, background): “The counterpart of this result for European option has been known for some time and lies at the basis of the fruitful interaction of robust finance and (martingale) optimal transport in the last decade, see [41, 33, 40, 29, 22, 19, 37, 26] among many others.”
Structure of optimal martingale transport plans in general dimensions (2015, background): “See for example Hobson [18], Beiglböck-Cox-Huesmann [6] and Beiglböck-Henry-Labordere-Touzi [4].”
Monotone martingale transport plans and Skorokhod embedding (2017, unknown): “2) have been provided in [30, 3, 6], see [2] for a variant applicable to Skorokhod problem.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, background): “[17, 30, 49]) and the Skorokhod embedding problem [7, 35].”
Existence, duality, and cyclical monotonicity for weak transport costs (2018, background): “More recently, variants of this ‘monotonicity priniciple’ have been applied in transport problems for finitely or infinitely many marginals [32, 18, 26, 11, 44], the martingale version of the optimal transport problem [12, 31, 13], the Skorokhod embedding problem [9] and the distribution constrained”
All adapted topologies are equal (2019, unknown): “[34, 13, 27, 23, 17, 33, 18, 12, 14] among many others).”
Optimal Skorokhod embedding given full marginals and Azema-Yor peacocks (2015, background): “king the limit of a duality result for a general optimal SEP under ﬁnitely-many marginal constraints, established in Guo, Tan and Touzi [12] (extending a duality result in Beiglbock, Cox and Huesmann [3]), we obtain a general duality result for the optimal SEP under full marginal constraints. Therea”