Suppose I have a Brownian motion $X_t$ with $X_0=0$. Let $T$ be the first exit time of the interval $[-1,1]$.
I'm trying to get a "quick" lower bound for the probability that $T$ is very large which is asymptotically reasonable. It's very easy to come up with nice upper bounds, but I can't find a way of bounding it below.
I can get something with the reflection principle, but it's inelegant. Is there a trick or a pretty way of doing it.
Best Answer
Note that $\mathbb E[\mathrm e^{sT}]=1/\cos(\sqrt{2s})$ for every $0\leqslant s\lt\pi^2/8$ and that $1/\cos(x)\to+\infty$ when $x\to\pi/2$, $x\lt\pi/2$. Thus, for every $a\lt\pi^2/8\lt b$ there exists some finite $C_a$ and $C_b$ such that, for every $t\geqslant0$, $$ C_b-bt\leqslant\log \mathbb P[T\geqslant t]\leqslant C_a-at. $$