Asymptotic distribution of the derivative of the taut string accompanying Wiener process

TL;DR

This paper derives the asymptotic distribution of the derivative of the taut string associated with a Wiener process, providing explicit formulas for minimal energy in a fixed-width strip over long time intervals.

Contribution

It introduces the asymptotic distribution of the derivative of the taut string for Wiener processes and derives explicit minimal energy expressions.

Findings

Minimal energy per unit time tends to π²/6r² for kinetic energy.

Explicit formulas for minimal energy of absolutely continuous functions.

Asymptotic distribution results applicable to long time intervals.

Abstract

In the article, we find the asymptotic distribution of the derivative of the taut string accompanying a Wiener process in a strip of fixed width on long time intervals. This enables to find explicit expressions for minimal energy (averaged function of the derivative) of an absolutely continuous function in this strip. For example, for kinetic energy which was considered earlier by Lifshits and Setterqvist, the minimal energy per unit of time tends to $\pi^2/6r^2$ where $r$ is the strip width.