The Gradient Flow of the Bass Functional in Martingale Optimal Transport

Paper: arXiv 2407.18781

Authors: Julio Backhoff-Veraguas, Gudmund Pammer, Walter Schachermayer

Abstract

Given $μ$ and $ν$, probability measures on $\mathbb R^d$ in convex order, a Bass martingale is arguably the most natural martingale starting with law $μ$ and finishing with law $ν$. Indeed, this martingale is obtained by stretching a reference Brownian motion so as to meet the data $μ,ν$. Unless $μ$ is a Dirac, the existence of a Bass martingale is a delicate subject, since for instance the reference Brownian motion must be allowed to have a non-trivial initial distribution $α$, not known in advance. Thus the key to obtaining the Bass martingale, theoretically as well as practically, lies in finding $α$. In \cite{BaSchTsch23} it has been shown that $α$ is determined as the minimizer of the so-called Bass functional. In the present paper we propose to minimize this functional by following its gradient flow, or more precisely, the gradient flow of its $L^2$-lift. In our main result we show that this gradient flow converges in norm to a minimizer of the Bass functional, and when $d=1$ we further establish that convergence is exponentially fast.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly rigorous theoretical analysis of the gradient flow of the Bass functional in martingale optimal transport, establishing existence, uniqueness, and norm convergence (exponential in 1D). However, it is entirely theoretical with no empirical backtests, computational implementation, or real financial market data.

Research Flowchart

  flowchart TD
    A[Goal: Find initial distribution α to construct Bass Martingale between measures μ and ν] --> B[Input Data: Probability measures μ, ν on ℝᵈ in convex order]
    B --> C[Methodology: Minimize Bass Functional via L²-lift Gradient Flow]
    C --> D[Computational Process: Track time-continuous evolution of measure α_t]
    D --> E[Outcome 1: Prove norm convergence to Bass Functional minimizer for general d]
    D --> F[Outcome 2: Prove exponentially fast convergence rate when d = 1]

Cited References (17)

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3. Dispersion-constrained martingale Schr "o dinger problems and the exact joint S&P 500/VIX smile calibration puzzle (2024)
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8. The Measure Preserving Martingale Sinkhorn Algorithm (2023)

4 of 17 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.