A note on the range of stochastic processes

A Brownian motion with drift is simply a process V t eta of the form V (eta) (t) = B (t) + eta t where B (t) is a standard Brownian motion and eta > 0. In [7], the authors showed that the underlying range R- t ( V- eta ) = sup(0) <=( s) <= (t) V- t(eta) is equivalent to eta t a.e in the long run, i.e Rt ( V-t (eta)) / t a.e -> t ->infinity eta. (0.1) In this paper, we show that (0.1) follows from a deterministic property. More precisely, we show that the long run behavior of the range of a (deterministic) function is obtainable straightaway from that of the function itself.