Stretched Brownian Motion: convergence of dual optimising sequences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914009407553536 |
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| author | Schachermayer, Walter Siorpaes, Pietro |
| author_facet | Schachermayer, Walter Siorpaes, Pietro |
| contents | We consider an irreducible pair $μ\leq_c ν$ of probability measures on $\mathbb{R}^d$ in convex order. In arXiv:2306.11019, Backhoff, Beiglböck, Schachermayer and Tschiderer have shown that the Stretched Brownian Motion from $μ$ to $ν$ is a Bass martingale, that there exists a dual optimiser $ψ_{lim}$, and the following somewhat surprising convergence result: by adding affine functions, one can make any dual optimising sequence $(ψ_n)_n$ (satisfying some minor technical conditions) converge pointwise to $ψ_{lim}$, save possibly on the relative boundary of the convex hull of the support of $ν$. In the present paper we deal with the more delicate issue of convergence on said boundary, showing in particular that $ψ_{lim}$ is $ν$ a.s. finite, and $(ψ_n)_n$ converges to $ψ_{lim}$ in $ν$-measure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_20017 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stretched Brownian Motion: convergence of dual optimising sequences Schachermayer, Walter Siorpaes, Pietro Probability Primary 60G42, 60G44, Secondary 91G20 We consider an irreducible pair $μ\leq_c ν$ of probability measures on $\mathbb{R}^d$ in convex order. In arXiv:2306.11019, Backhoff, Beiglböck, Schachermayer and Tschiderer have shown that the Stretched Brownian Motion from $μ$ to $ν$ is a Bass martingale, that there exists a dual optimiser $ψ_{lim}$, and the following somewhat surprising convergence result: by adding affine functions, one can make any dual optimising sequence $(ψ_n)_n$ (satisfying some minor technical conditions) converge pointwise to $ψ_{lim}$, save possibly on the relative boundary of the convex hull of the support of $ν$. In the present paper we deal with the more delicate issue of convergence on said boundary, showing in particular that $ψ_{lim}$ is $ν$ a.s. finite, and $(ψ_n)_n$ converges to $ψ_{lim}$ in $ν$-measure. |
| title | Stretched Brownian Motion: convergence of dual optimising sequences |
| topic | Probability Primary 60G42, 60G44, Secondary 91G20 |
| url | https://arxiv.org/abs/2508.20017 |