Suppose $π$ is a one-dimensional standard Brownian motion defined on some probability space $(\Omega, \mathcal F, P)$ and let $π(π‘):=\exp\{π(π‘)β\frac{1}{2}π‘β\frac{1}{π‘+1}\}$ for $π‘\ge 0$. Note that $π(\infty):=\limsup_{t\to\infty}π(π‘)=0$ a.s. because $\lim_{t\to\infty}\frac{π(π‘)}{π‘}=0$ a.s.

My question is: How to show that $E[\sup_{0\le t\le \infty}π(π‘)]=\infty$? Many thanks. (I posed this question in stackexchange earlier today. Not sure if this is the right place to ask this question.)

Many thanks.