Dual attainment for the martingale transport problem

DOI10.3150/17-BEJ1015zbMATH Open1470.49071arXiv1705.04273OpenAlexW2962893500MaRDI QIDQ2419652

Jan Obłój, Tongseok Lim, Mathias Beiglböck

Publication date: 14 June 2019

Published in: Bernoulli (Search for Journal in Brave)

Abstract: We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [BNT16] established such existence for weak (quasi-sure) duality, [BHP13] showed existence for the natural stronger pointwise duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when $ymapstoc(x,y)$ is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost $c(x,y)$ is twice continuously differentiable in $y$. Further, this may not be improved as we give examples with $c(x,cdot)in{C}^{2-epsilon}$, $epsilon>0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.

Full work available at URL: https://arxiv.org/abs/1705.04273

zbMATH Keywords

Kantorovich dualitymartingale optimal transportdual attainmentrobust mathematical finance

Mathematics Subject Classification ID

Duality theory (optimization) (49N15) Martingales and classical analysis (60G46) Optimal transportation (49Q22) Financial markets (91G15)

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