Integrability and concentration of the truncated variation for the sample paths of fractional Brownian motions, diffusions and L\'{e}vy processes

TL;DR

This paper investigates the exponential integrability and concentration properties of truncated variation for fractional Brownian motions, diffusions, and Lévy processes, introducing new techniques and conditions for their probabilistic behavior.

Contribution

It develops a chaining-based technique to prove Gaussian concentration for diffusions and establishes conditions for exponential moments of Lévy process truncated variation.

Findings

Gaussian concentration for certain diffusions

Necessary and sufficient conditions for exponential moments of Lévy processes

New chaining approach for analyzing truncated variation

Abstract

For a real c\`{a}dl\`{a}g function $f$ defined on a compact interval, its truncated variation at the level $c>0$ is the infimum of total variations of functions uniformly approximating $f$ with accuracy $c/2$ and (in opposite to the total variation) is always finite. In this paper, we discuss exponential integrability and concentration properties of the truncated variation of fractional Brownian motions, diffusions and L\'{e}vy processes. We develop a special technique based on chaining approach and using it we prove Gaussian concentration of the truncated variation for certain class of diffusions. Further, we give sufficient and necessary condition for the existence of exponential moment of order $\alpha>0$ of truncated variation of L\'{e}vy process in terms of its L\'{e}vy triplet.