A tree approach to $p$-variation and to integration

TL;DR

This paper introduces a novel tree-based framework linking the geometric structure of paths to their $p$-variation and integrability, providing new insights and tools for analyzing stochastic processes.

Contribution

It develops a tree representation of paths that connects fractal dimension, $p$-variation, and integration, including applications to stochastic processes like Brownian motion and Lévy processes.

Findings

Fractal dimension of the tree is estimated from path variations.

Young and rough path integrals are expressed as integrals on the tree.

An estimator for the Hurst parameter of fractional Brownian motion is proposed.

Abstract

We consider a real-valued path; it is possible to associate a tree to this path, and we explore the relations between the tree, the properties of $p$-variation of the path, and integration with respect to the path. In particular, the fractal dimension of the tree is estimated from the variations of the path, and Young integrals with respect to the path, as well as integrals from the rough paths theory, are written as integrals on the tree. Examples include some stochastic paths such as martingales, L\'evy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given).