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

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}.