# Brownian motion estimate

• Jun 24th 2010, 10:13 AM
akbar
Brownian motion estimate
Let $W_t$ be a one-dimensional Brownian motion. How do you prove that one can find constants $c, \lambda >0$ such that:
$\mathbb{P}\Big(\sup_{1\leq s\leq 2^t}\frac{|W_s|}{\sqrt{s}}\leq 1\Big)\leq ce^{-\lambda t},\quad t\geq 1.$
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.

a trivial solution is available, since $P(anything) \leq 1$. Values of $c , \lambda$ can be chosen from that.