arXiv subjects

Armand Ley

Publications and source records attributed to Armand Ley.

3 recordsLinked to original sources

arXiv2026

Bridging classical and martingale Schr\"odinger bridges

We investigate the martingale Schr\"odinger bridge, recently introduced by Nutz and Wiesel as a distinguished martingale transport plan between two probability measures in convex order. We show that this construction extends naturally to arbitrary dimension and admits several equivalent characterizations. In particular, we identify its continuous-time counterpart as the continuous martingale with prescribed marginals that minimizes a weighted quadratic energy measuring the deviation from Brownian motion. In the irreducible case, we prove that this continuous martingale Schr\"odinger bridge coincides with the F\"ollmer martingale, that is, with the Doob martingale associated to a suitable F\"ollmer process. More generally, we relate the martingale Schr\"odinger bridge to a variational problem over base measures and to the dual formulation of the corresponding weak optimal transport problem, thereby clarifying its connection with the classical Schr\"odinger bridge.

math.PR↗

arXiv2025

Entropic selection for optimal transport on the line with distance cost

We study the small-regularisation limit of the entropic optimal transport problem on the line with distance cost. While convergence of entropic minimizers is well understood in the discrete setting and in the case where the cost is continuous and there is a unique optimal transport plan, the question of existence and characterization outside these settings remains largely open. We propose a natural candidate for the limiting object and establish its convergence under mutual singularity of the marginals. For arbitrary marginals, we moreover prove that every limit point of entropic minimizers obeys a structural condition known as weak multiplicativity. The construction of our candidate relies on a decomposition theorem for optimal transport plan that we believe is of independent interest. This article complements the previous work of Di Marino and Louet.

math.OC↗

arXiv2024

On the Markov transformation of Gaussian processes

Given a Gaussian process $(X_t)_{t \in \mathbb{R}}$, we construct a Gaussian \emph{Markov} process with the same one-dimensional marginals using sequences of transformations of $(X_t)_{t \in \mathbb{R}}$ "made Markov" at finitely many times. We prove that there exists at least such a Markov transform of $(X_t)_{t \in \mathbb{R}}$. In the case the instantaneous decorrelation rate of $(X_t)_{t \in \mathbb{R}}$ is continuous, we prove that the Markov transform is uniquely determined and characterized through the same instantaneous decorrelation rate.

math.PR↗