# Stretched Brownian Motion: convergence of dual optimising sequences

## TL;DR

This paper studies the convergence properties of dual optimising sequences in the context of Stretched Brownian Motion between probability measures in convex order, extending previous results to boundary points and establishing convergence in measure.

## Contribution

It advances understanding of dual optimisers by proving convergence on the boundary of the support, and shows the finiteness of the limit dual function almost surely.

## Key findings

- Dual optimising sequences converge in measure on the boundary.
- The limit dual function is finite almost surely.
- Convergence extends previous pointwise results to boundary points.

## Abstract

We consider an irreducible pair $\mu \leq_c \nu$ of probability measures on $\mathbb{R}^d$ in convex order. In arXiv:2306.11019, Backhoff, Beiglb\"ock, Schachermayer and Tschiderer have shown that the Stretched Brownian Motion from $\mu$ to $\nu$ is a Bass martingale, that there exists a dual optimiser $\psi_{lim}$, and the following somewhat surprising convergence result: by adding affine functions, one can make any dual optimising sequence $(\psi_n)_n$ (satisfying some minor technical conditions) converge pointwise to $\psi_{lim}$, save possibly on the relative boundary of the convex hull of the support of $\nu$. In the present paper we deal with the more delicate issue of convergence on said boundary, showing in particular that $\psi_{lim}$ is $\nu$ a.s. finite, and $(\psi_n)_n$ converges to $\psi_{lim}$ in $\nu$-measure.