A note on a deterministic property to obtain the long run behavior of the range of a stochastic process

TL;DR

This paper demonstrates that the long-term behavior of the range of a stochastic process with drift can be derived from a deterministic property of the process, providing a new perspective on asymptotic analysis.

Contribution

It introduces a deterministic property that directly yields the asymptotic behavior of the range of processes with drift, extending previous stochastic results.

Findings

Range of drifted Brownian motion asymptotically equals drift times time

Deterministic property can determine long-term range behavior

Provides a continuous analogue of a known discrete result

Abstract

A Brownian motion with drift is simply a process $V^{\eta}_t$ of the form $V^{\eta}_t=B_{t}+\eta t$ where $B_{t}$ is a standard Brownian motion and $\eta>0$ \footnote{The case $\eta<0$ is deducible by remarking $V^{-\eta}(t)=-V^{\eta}(t)$.} In \cite{tanre2006range}, the authors considered the drifted Brownian motion and studied the statistics of some related sequences defined by certain stopping times. In particular, they provided the law of the range $R_{t}(V^{\eta})$ of $V^{\eta}$ as well as its first range process $\theta_{V^{\eta}}(a)$. In particular, they investigated the asymptotic comportment of $R_{t}(V^{\eta})$ and $\theta_{V^{\eta}}(a)$. They proved that if $V_{t}^{\eta}$ is a Brownian motion with a positive drift $\eta$ then its range $R_{t}(V^{\eta})=\sup_{0\leq s\leq t}V_{t}^{\eta}-\inf_{0\leq s\leq t}V_{t}^{\eta}$ is asymptotically equivalent to $\eta t$. In other words \begin{equation} \frac{R_{t}(V^{\eta})}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}\eta.\label{range} \end{equation} In this paper, we show that (\ref{range}) follows from a striking deterministic property. More precisely, we show that the long run behavior of the range of a deterministic continuous function is obtainable straightaway from that of the function itself. Our result can be deemed as the continuous version of a similar one appeared in \cite{mgrw}.