Expected Regularized Total Variation of Brownian Motion

TL;DR

This paper defines a new regularized total variation for continuous functions, constructs an explicit partition to achieve it, and estimates its expected value for Brownian motion, providing a novel way to analyze functions with unbounded variation.

Contribution

It introduces a regularized total variation concept, offers an explicit partition construction, and estimates its expectation specifically for Brownian motion.

Findings

Explicit partition construction for regularized total variation

Estimated expected regularized total variation of Brownian motion

Provides a new framework for analyzing unbounded variation functions

Abstract

We introduce a notion of regularized total variation on an interval for continuous functions with unbounded variation. The definition of regularized total variation is obtained from that of total variation by subtracting a penalty for the size of the partition used to estimate the variation. We present an explicit construction of a partition achieving the regularized total variation, and use this construction to estimate the expected regularized total variation of Brownian motion on an interval.