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In this article, we characterize solutions to the martingale Benamou$-$Brenier problem as $\textit{Bass martingales}$, i.e. transformations of Brownian motion through the gradient of a convex function. Our result is based on a new (static) Brenier-type theorem for a particular weak martingale optimal transport problem. As in the classical case, the structure of the primal optimizer is derived from its dual counterpart, whose derivation forms the technical core of this article. A key challenge is that dual attainment is a subtle issue in martingale optimal transport, where dual optimizers may fail to exist, even in highly regular settings.
From: Bertram Tschiderer [view email]
[v1]
Mon, 19 Jun 2023 15:31:15 UTC (45 KB)
[v2]
Wed, 2 Oct 2024 08:54:06 UTC (46 KB)
[v3]
Sun, 30 Mar 2025 15:34:03 UTC (620 KB)
[v4]
Fri, 10 Jul 2026 16:46:13 UTC (614 KB)
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