Complete Duality for Martingale Optimal Transport on the Line

By Mathias Beiglbock, Marcel Nutz, Nizar Touzi. 2015

146 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/68e725a0104852a4fd39ec74ea3f76c1f088c172

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Citations in the corpus

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”
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.”
Martingale Benamou–Brenier: A probabilistic perspective (2017, result): “Recently a number of similar results have been obtained in related areas, see [15, 48, 35, 12, 66, 37, 16, 56, 57, 17].”
Irreducible convex paving for decomposition of multidimensional martingale transport plans (2017, methodology): “2 in [4], indeed in dimension one J “ Jθ “ J̄ by Proposition 2.”
A new class of costs for optimal transport planning (2018, methodology): “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.”
Exponential utility maximization under model uncertainty for unbounded endowments (2016, background): “probabilistic model and the consideration of a whole family of models as a consequence. This is not a new concept and since the seminal papers [16] and [24] it has gained a lot of attention, see e.g. [1, 3, 4, 6, 8, 12, 18, 28, 29, 30] and [9] for an overview. To state our problem more precisely, gi”