Bi-martingale optimal transport and its applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909880258920448 |
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| author | Bołbotowski, Karol |
| author_facet | Bołbotowski, Karol |
| contents | We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale framework underlies and interconnects several variational problems on the space of probability measures. For the quadratic cost, it provides an optimal transport interpretation of the second Zolotarev distance on $\mathrm{P}_2(\mathbb{R}^d)$. For a broader class of convex costs, it leads to optimization problems under convex order constraints, encompassing in particular the Zolotarev projection onto the cone of dominating probability measures. As a main application, we construct a $Γ$-convergent bi-martingale approximation of the classical martingale optimal transport problem. This scheme robustly accommodates deviations from convex order between the marginal distributions and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_27451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bi-martingale optimal transport and its applications Bołbotowski, Karol Probability Optimization and Control 49Q22, 60E15, 60G42, 49J55, 60A10 We introduce a new non-linear optimal transport formulation for a pair of probability measures on $\mathbb{R}^d$ sharing a common barycentre, in which admissible transference plans satisfy two martingale-type constraints. This bi-martingale framework underlies and interconnects several variational problems on the space of probability measures. For the quadratic cost, it provides an optimal transport interpretation of the second Zolotarev distance on $\mathrm{P}_2(\mathbb{R}^d)$. For a broader class of convex costs, it leads to optimization problems under convex order constraints, encompassing in particular the Zolotarev projection onto the cone of dominating probability measures. As a main application, we construct a $Γ$-convergent bi-martingale approximation of the classical martingale optimal transport problem. This scheme robustly accommodates deviations from convex order between the marginal distributions and overcomes the well-known instability of MOT with respect to variations of the marginals in higher dimensions. |
| title | Bi-martingale optimal transport and its applications |
| topic | Probability Optimization and Control 49Q22, 60E15, 60G42, 49J55, 60A10 |
| url | https://arxiv.org/abs/2510.27451 |