The mean of a running maximum

Question

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

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

Many thanks.

Answer 1

This seems to me a standard exercice in a probability course: Ignore the $1/(t+1)$ term as $X(t)\geq e^{-1}\exp(W(t)-t/2)$. The term $M_t:=\exp(W(t)-t/2)$ is well known to be a martingale so $\mathbb{E}(M_t)=1$ for all $t$. If $\mathbb{E}(\sup_{0\leq t\leq \infty}X_t)<\infty$ then by dominated convergence $$1=\mathbb{E}(M_t)\rightarrow_{t\rightarrow \infty} \mathbb{E}(M_\infty)=0$$