I'm trying to solve a problem: $$ \sup_{0 \leq \tau \leq 1} E W_{\tau - \varepsilon}, $$ where $W$ is a Wiener process and $\varepsilon$ is a fixed real number.
I've tried to approximate a Wiener process by a random walk, but I couldn't get any good results (maybe, I missed something and that's a good idea). So, what think you about that problem? Any ideas, hints, or even a solution