Let $\displaystyle W_t$ be a one-dimensional Brownian motion. How do you prove that one can find constants $\displaystyle c, \lambda >0$ such that:

$\displaystyle \mathbb{P}\Big(\sup_{1\leq s\leq 2^t}\frac{|W_s|}{\sqrt{s}}\leq 1\Big)\leq ce^{-\lambda t},\quad t\geq 1.$One usual approach to prove such inequalities, using stopping times along the lines of the proof of the Maximal Theorem, doesn't seem to quite work here.

By advance, thank you very much for your help.