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Existence of Bass martingales and the martingale Benamou$-$Brenier problem in $\mathbb{R}^{d}$

Julio Backhoff-Veraguas, Mathias Beiglböck, Walter Schachermayer, Bertram Tschiderer

TL;DR

The paper addresses the martingale Benamou–Brenier problem in $\mathbb{R}^d$ by showing that SBM (stretched Brownian motion) optimizers are Bass martingales, i.e., $M_t=\mathbb{E}[\nabla v(B_1)\mid \mathcal{F}_t]$ for a convex $v$. It develops a static Brenier-type duality for a weak martingale transport problem and proves that there is no duality gap; under irreducibility a dual optimizer exists and gives rise to a Bass martingale, linking the primal optimizer to a Brenier-type map. The results provide a complete structural bridge between continuous-time martingale transport and gradient-transformations of Brownian motion, yielding a time-consistent interpolation $\mu_t=\mathcal{L}(M_t)$ between $\mu$ and $\nu$ and a sharp characterization of when Bass martingales exist. These findings extend classical Brenier/McCann theory to the martingale setting, with implications for Skorokhod embedding and finance where martingale constraints are natural.

Abstract

In classical optimal transport, the contributions of Benamou$-$Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. 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.

Existence of Bass martingales and the martingale Benamou$-$Brenier problem in $\mathbb{R}^{d}$

TL;DR

The paper addresses the martingale Benamou–Brenier problem in by showing that SBM (stretched Brownian motion) optimizers are Bass martingales, i.e., for a convex . It develops a static Brenier-type duality for a weak martingale transport problem and proves that there is no duality gap; under irreducibility a dual optimizer exists and gives rise to a Bass martingale, linking the primal optimizer to a Brenier-type map. The results provide a complete structural bridge between continuous-time martingale transport and gradient-transformations of Brownian motion, yielding a time-consistent interpolation between and and a sharp characterization of when Bass martingales exist. These findings extend classical Brenier/McCann theory to the martingale setting, with implications for Skorokhod embedding and finance where martingale constraints are natural.

Abstract

In classical optimal transport, the contributions of BenamouBrenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. In this article, we characterize solutions to the martingale BenamouBrenier problem as , 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.
Paper Structure (26 sections, 37 theorems, 217 equations, 4 figures)

This paper contains 26 sections, 37 theorems, 217 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Benamou--Brenier transport problem and McCann interpolation in probabilistic terms
  3. Martingale optimization problem
  4. The Bass martingale
  5. Structure of stretched Brownian motion
  6. Weak transport formulation and dual viewpoint
  7. Bass martingales for specific marginals
  8. Literature
  9. Structure of the paper
  10. Definitions and notation
  11. Duality for stretched Brownian motion
  12. Proof of the first part of Theorem 1.4
  13. Preparation for the proof of the second part of Theorem 1.4
  14. Existence of dual optimizers and Bass martingales
  15. Irreducibility and existence of dual optimizers
  16. ...and 11 more sections

Key Result

Theorem 1.3

Let $\mu \preceq_{\textnormal{c}} \nu$ be probabilities on $\mathbb{R}^{d}$ with finite second moments and suppose that $(\mu,\nu)$ is irreducible. Then the following are equivalent for a martingale $M = (M_{t})_{0 \leqslant t \leqslant 1}$ with $M_{0} \sim \mu$ and $M_{1} \sim \nu$:

Figures (4)

  • Figure 1: Sample paths of $M$, different starting values.
  • Figure 2: Sample paths of $M$, identical starting value.
  • Figure 3: There is a martingale which starts uniformly distributed on the middle circle ($r=1$), terminates with equal probability on the inner circle ($r=1/2$) or outer circle ($r=8/5$), and moves only on the violet line segments. The angle between the segments and the middle circle is chosen so that the lengths to the inner and outer circle are equal.
  • Figure 4: Dependencies between results.

Theorems & Definitions (86)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • Proposition 3.5
  • proof
  • ...and 76 more