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

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, Lévy processes and fractional Brownian motions (for which an estimator of the Hurst parameter is given).

References in corpus (2)

• The coding of compact real trees by real valued functions
• Local independence of fractional Brownian motion

Cited by in corpus (5)

• The Play Operator, the Truncated Variation and the Generalisation of the Jordan Decomposition
• Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and Lévy processes
• $\zeta$-functions and the topology of superlevel sets of stochastic processes
• On $C^0$-persistent homology and trees
• Moments approach to the concentration properties of truncated variation