On limit distributions of normalized truncated variation, upward truncated variation and downward truncated variation processes

TL;DR

This paper introduces and analyzes the properties of truncated variation processes for Brownian motion with drift, establishing their convergence and approximation behaviors, and providing tools for studying continuous stochastic processes.

Contribution

It defines truncated variation concepts and proves their convergence and approximation properties for Brownian motion, offering a robust methodology applicable to various continuous processes.

Findings

Truncated variation approximates deterministic processes as the truncation parameter tends to zero.

Error in approximation converges weakly to a Brownian motion.

Results include analogs of classical theorems like Anscombe-Donsker and Laplace transforms for drawdowns.

Abstract

In the paper we introduce the truncated variation, upward truncated variation and downward truncated variation. These are closely related to the total variation but are well-defined even if the latter is infinite. Our aim is to explore their feasibility to studies of stochastic processes. We concentrate on a Brownian motion with drift for which we prove the convergence of the above- mentioned quantities. For example, we study the truncated variation when the truncation parameter c tends to 0. We prove in this case that for "small" c's it is well-approximated by a deterministic process. Moreover we prove that error in this approximation converges weakly (in functional sense) to a Brownian motion. We prove also similar result for truncated variation processes when time parameter is rescaled to infinity. We stress that our methodology is robust. A key to the proofs was a decomposition of the truncated variation (see Lemmas 11 and 12). It can be used for studies of any continuous processes. Some additional results like an analog of the Anscombe-Donsker theorem and the Laplace transform of time to given drawdown by c (and analogously drawup till time) are presented.