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 $V_t^{\eta }$ of the form $V_t^{\eta }=B_t+\eta t$ 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 $R_t(V^{\eta })={\sup }_{0\le s\le t}{V}_t^{\eta }-{\inf }_{0\le s\le t}{V}_t^{\eta }$ is asymptotically equivalent to $\eta t$. In other words \begin{equation} \frac{R_{t}(V^η)}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}η.\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}.