Energy of taut strings accompanying Wiener process

TL;DR

This paper investigates the asymptotic energy of taut strings around Wiener processes, providing bounds, simulations, and an adaptive pursuit strategy linked to Fisher information minimization.

Contribution

It introduces bounds and simulations for the energy of taut strings over large intervals and proposes an adaptive pursuit method based on past Wiener process values.

Findings

Energy of taut strings scales as C^2 T / r^2 for large T

An adaptive pursuit strategy minimizes asymptotic energy using only past data

The pursuit strategy relates to Fisher information minimization

Abstract

Let $W$ be a Wiener process. The function $h(\cdot)$ minmizing energy $\int_0^T h'(t)^2\, dt$ among all functions satisfying $W(t)-r \le h(t) \le W(t)+ r$ on an interval $[0,T]$ is called taut string. This is a classical object well known in Variational Calculus, Mathematical Statistics, etc. We show that the energy of this taut string on large intervals is equivalent to $C^2 T\, /\, r^2$ where $C$ is some finite positive constant. While the precise value of $C$ remains unknown, we give various theoretical bounds for it as well as rather precise results of computer simulation. While the taut string clearly depends on entire trajectory of $W$, we also consider an adaptive version of the problem by giving a construction (Markovian pursuit) of a random function based only on the past values of $W$ and having minimal asymptotic energy. The solution, an optimal pursuit strategy, quite surprisingly turns out to be related with a classical minimization problem for Fisher information on the bounded interval.