Stretched Brownian Motion: convergence of dual optimising sequences

2025/08/27 by Schachermayer, Walter, Siorpaes, Pietro
#60G44 #FOS: Mathematics #Primary 60G42 #Probability (math.PR) #Secondary 91G20

paper · doi:10.48550/arxiv.2508.20017

Abstract

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

Citations

• Robust and Fast Bass Local Volatility
• The Gradient Flow of the Bass Functional in Martingale Optimal Transport
• The decomposition of stretched Brownian motion into Bass martingales
• Calibration of the Bass Local Volatility model
• The Bass functional of martingale transport
• Martingale Benamou--Brenier: a probabilistic perspective
• A Benamou-Brenier formulation of martingale optimal transport
• Local Variance Gamma and Explicit Calibration to Option Prices

Related

• The decomposition of stretched Brownian motion into Bass martingales
• Existence of Bass martingales and the martingale Benamou-Brenier problem in ℝ^d
• The Gradient Flow of the Bass Functional in Martingale Optimal Transport
• Structure of optimal martingale transport plans in general dimensions
• Dual attainment for the martingale transport problem