2024/07/26 by Julio Backhoff‐Veraguas, Backhoff-Veraguas, Julio, Gudmund Pammer +3 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2407.18781
openalex publication_date 2024/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given μ and ν, probability measures on \mathbb Rd 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 \citeBaSchTsch23 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 L2-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.