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Geometric Martingale Benamou-Brenier transport and geometric Bass martingales

Julio Backhoff, Gregoire Loeper, Jan Obloj

TL;DR

The paper develops a geometric analogue of Bass martingales by formulating a geometric martingale Benamou–Brenier problem and establishing a 1–1 correspondence with the arithmetic Bass martingale problem through a transform of marginals. It provides an explicit representation and SDE description of the geometric optimizer, a Kantorovich dual, and both probabilistic and PDE-based proofs, including a change of measure argument that links geometric and arithmetic solutions. A detailed analysis of the structure of the geometric Bass martingale is given, highlighting the Bass-type representation on irreducible components and a practical distributional formula for S. The work also clarifies the relationship between arithmetic and geometric Bass martingales, shows that geometric Brownian motion is the unique process that is simultaneously both, and introduces Martingale Sinkhorn systems as a numerical framework for solving these problems. These results offer a unified OT calibration framework for lognormal-type dynamics with potential finance applications and provide efficient numerical routes via Sinkhorn-inspired fixed point iterations.

Abstract

We introduce and study geometric Bass martingales. Bass martingales were introduced in \cite{Ba83} and studied recently in a series of works, including \cite{BaBeHuKa20,BaBeScTs23}, where they appear as solutions to the martingale version of the Benamou-Brenier optimal transport formulation. These arithmetic, as well as our novel geometric, Bass martingales are continuous martingale on $[0,1]$ with prescribed initial and terminal distributions. An arithmetic Bass martingale is the one closest to Brownian motion: its quadratic variation is as close as possible to being linear in the averaged $L^2$ sense. Its geometric counterpart we develop here, is the one closest to a geometric Brownian motion: the quadratic variation of its logarithm is as close as possible to being linear. By analogy between Bachelier and Black-Scholes models in mathematical finance, the newly obtained geometric Bass martingales} have the potential to be of more practical importance in a number of applications. Our main contribution is to exhibit an explicit bijection between geometric Bass martingales and their arithmetic counterparts. This allows us, in particular, to translate fine properties of the latter into the new geometric setting. We obtain an explicit representation for a geometric Bass martingale for given initial and terminal marginals, we characterise it as a solution to an SDE, and we show that geometric Brownian motion is the only process which is both an arithmetic and a geometric Bass martingale. Finally, we deduce a dual formulation for our geometric martingale Benamou-Brenier problem. Our main proof is probabilistic in nature and uses a suitable change of measure, but we also provide PDE arguments relying on the dual formulation of the problem, which offer a rigorous proof under suitable regularity assumptions.

Geometric Martingale Benamou-Brenier transport and geometric Bass martingales

TL;DR

The paper develops a geometric analogue of Bass martingales by formulating a geometric martingale Benamou–Brenier problem and establishing a 1–1 correspondence with the arithmetic Bass martingale problem through a transform of marginals. It provides an explicit representation and SDE description of the geometric optimizer, a Kantorovich dual, and both probabilistic and PDE-based proofs, including a change of measure argument that links geometric and arithmetic solutions. A detailed analysis of the structure of the geometric Bass martingale is given, highlighting the Bass-type representation on irreducible components and a practical distributional formula for S. The work also clarifies the relationship between arithmetic and geometric Bass martingales, shows that geometric Brownian motion is the unique process that is simultaneously both, and introduces Martingale Sinkhorn systems as a numerical framework for solving these problems. These results offer a unified OT calibration framework for lognormal-type dynamics with potential finance applications and provide efficient numerical routes via Sinkhorn-inspired fixed point iterations.

Abstract

We introduce and study geometric Bass martingales. Bass martingales were introduced in \cite{Ba83} and studied recently in a series of works, including \cite{BaBeHuKa20,BaBeScTs23}, where they appear as solutions to the martingale version of the Benamou-Brenier optimal transport formulation. These arithmetic, as well as our novel geometric, Bass martingales are continuous martingale on with prescribed initial and terminal distributions. An arithmetic Bass martingale is the one closest to Brownian motion: its quadratic variation is as close as possible to being linear in the averaged sense. Its geometric counterpart we develop here, is the one closest to a geometric Brownian motion: the quadratic variation of its logarithm is as close as possible to being linear. By analogy between Bachelier and Black-Scholes models in mathematical finance, the newly obtained geometric Bass martingales} have the potential to be of more practical importance in a number of applications. Our main contribution is to exhibit an explicit bijection between geometric Bass martingales and their arithmetic counterparts. This allows us, in particular, to translate fine properties of the latter into the new geometric setting. We obtain an explicit representation for a geometric Bass martingale for given initial and terminal marginals, we characterise it as a solution to an SDE, and we show that geometric Brownian motion is the only process which is both an arithmetic and a geometric Bass martingale. Finally, we deduce a dual formulation for our geometric martingale Benamou-Brenier problem. Our main proof is probabilistic in nature and uses a suitable change of measure, but we also provide PDE arguments relying on the dual formulation of the problem, which offer a rigorous proof under suitable regularity assumptions.
Paper Structure (8 sections, 8 theorems, 61 equations)

This paper contains 8 sections, 8 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Kantorovitch duality perspective on geometric Bass martingale
  5. A probabilistic proof of Theorem \ref{['thm:g-a equiv']} via a change of measure
  6. Structure of the geometric Bass martingale
  7. Relations between classes of Bass martingales
  8. Martingale Sinkhorn systems

Key Result

Lemma 2.1

Let $\mu_0,\mu_1\in \mathcal{P}_{-1,1}({\mathbb R}_+)$ and $\nu_i:=Id_\dag\mu_i$, $i=0,1$. Then $\nu_0,\nu_1\in \mathcal{P}_{0,2}({\mathbb R}_+)$ and In case $\mu_0\preccurlyeq_{cx}\mu_1$, we have $I\in \mathcal{I}_{[\nu_0,\nu_1]} \iff \left\{\frac{1}{x}:x\in I\right\}\in\mathcal{I}_{[\mu_0,\mu_1]},$ and conversely $J\in \mathcal{I}_{[\mu_0,\mu_1]} \iff \left\{\frac{1}{x}:x\in J\right\}\in\mathca

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Remark 3.4
  • Proposition 4.1
  • Lemma 4.2
  • proof : Proof of Theorem \ref{['thm:g-a equiv']}
  • Remark 5.1
  • ...and 4 more