Multiperiod martingale transport

DOI10.1016/J.SPA.2019.05.010zbMATH Open1444.60033arXiv1703.10588OpenAlexW2604253913WikidataQ127759380 ScholiaQ127759380MaRDI QIDQ2301489

Florian Stebegg, Marcel Nutz, Xiaowei Tan

Publication date: 24 February 2020

Published in: Stochastic Processes and their Applications (Search for Journal in Brave)

Abstract: Consider a multiperiod optimal transport problem where distributions $m{u}_0,dots,m{u}_n$ are prescribed and a transport corresponds to a scalar martingale $X$ with marginals $X_{t}simm{u}_t$. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence--Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if $m{u}_0$ is atomless, but not in general. In the one-period case $n=1$, these transports reduce to the Left-Curtain coupling of Beiglb"ock and Juillet. In the multiperiod case, the bivariate marginals for dates $(0,t)$ are of Left-Curtain type, if and only if $m{u}_0,dots,m{u}_n$ have a specific order property. The general analysis of the transport problem also gives rise to a strong duality result and a description of its polar sets. Finally, we study a variant where the intermediate marginals $m{u}_1,dots,m{u}_{n-1}$ are not prescribed.

Full work available at URL: https://arxiv.org/abs/1703.10588

zbMATH Keywords

dualityoptimal transportmartingale coupling

Mathematics Subject Classification ID

Martingales with discrete parameter (60G42) Linear optimal control problems (49N05)

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Related Items (14)

Fine properties of the optimal Skorokhod embedding problem ⋮ The geometry of multi-marginal Skorokhod embedding ⋮ Shadows and barriers ⋮ Supermartingale Brenier's theorem with full-marginals constraint ⋮ Supermartingale shadow couplings: the decreasing case ⋮ A potential-based construction of the increasing supermartingale coupling ⋮ Martingale Benamou-Brenier: a probabilistic perspective ⋮ ROBUST BOUNDS FOR DERIVATIVE PRICES IN MARKOVIAN MODELS ⋮ Change of numeraire in the two-marginals martingale transport problem ⋮ Martingale optimal transport in the discrete case via simple linear programming techniques ⋮ Connecting GANs, mean-field games, and optimal transport ⋮ Shadow couplings ⋮ Shadow martingales -- a stochastic mass transport approach to the peacock problem ⋮ The potential of the shadow measure

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