On truncated variation, upward truncated variation and downward truncated variation for diffusions

TL;DR

This paper investigates the properties of truncated variation, especially for diffusions, revealing its connections to quadratic variation and establishing convergence results, thus offering new tools for stochastic process analysis.

Contribution

It introduces and analyzes upward and downward truncated variations for diffusions, establishing their asymptotic behavior and connections to quadratic variation and Ocone martingales.

Findings

Normalized truncated variation converges to quadratic variation for semimartingales.

For diffusions, truncated variation converges weakly to an Ocone martingale.

Truncated variation is always finite and approximates total variation within a threshold.

Abstract

The truncated variation, $TV^c$, is a fairly new concept introduced in [5]. Roughly speaking, given a c\`adl\`ag function $f$, its truncated variation is "the total variation which does not pay attention to small changes of $f$, below some threshold $c>0$". The very basic consequence of such approach is that contrary to the total variation, $TV^c$ is always finite. This is appealing to the stochastic analysis where so-far large classes of processes, like semimartingales or diffusions, could not be studied with the total variation. Recently in [6], another characterization of $TV^c$ was found. Namely $TV^c$ is the smallest total variation of a function which approximates $f$ uniformly with accuracy $c/2$. Due to these properties we envisage that $TV^c$ might be a useful concept to the theory of processes. For this reason we determine some properties of $TV^c$ for some well-known processes. In course of our research we discover intimate connections with already known concepts of the stochastic processes theory. Firstly, for semimartingales we proved that $TV^c$ is of order $c^{-1}$ and the normalized truncated variation converges almost surely to the quadratic variation of the semimartingale as $c\searrow0$. Secondly, we studied the rate of this convergence. As this task was much more demanding we narrowed to the class of diffusions (with some mild additional assumptions). We obtained the weak convergence to a so-called Ocone martingale. These results can be viewed as some kind of large numbers theorem and the corresponding central limit theorem. All the results above were obtained in a functional setting, viz. we worked with processes describing the growth of the truncated variation in time. Moreover, in the same respect we also treated two closely related quantities - the so-called upward truncated variation and downward truncated variation.