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Beiglböck, Mathias; Lim, Tongseok; Obłój, Jan

Dual attainment for the martingale transport problem. (English) Zbl 1470.49071

Bernoulli 25, No. 3, 1640-1658 (2019).
Summary: We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While the first author et al. established such existence for weak (quasi-sure) duality [Ann. Probab. 45, No. 5, 3038–3074 (2017; Zbl 1417.60032)], in [Finance Stoch. 17, No. 3, 477–501 (2013; Zbl 1277.91162)], the first author et al. showed existence for the natural stronger (pointwise) duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when \(y\mapsto c(x,y)\) 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 \(c(x,\cdot)\in C^{2-\varepsilon}\), \(\varepsilon>0\), where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if \(c\) is Lipschitz.

Cited in 15 Documents

MSC:

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

Keywords:

dual attainment; Kantorovich duality; martingale optimal transport; robust mathematical finance

Citations:

Zbl 1417.60032; Zbl 1277.91162
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Full Text: DOI arXiv Euclid

References:

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