HardStochastic Processes

Expected Running Max of BM

Goldman SachsDE Shaw

For a standard Brownian motion, what is the expected value of its running maximum over [0, 1], i.e. $\mathbb{E}[\max_{0\le t\le 1} B_t]$ ? (Round to 3 decimals.)

Approach

Reflection principle: the running max is distributed like $|B_t|$ .

$E[|Z|] = \sqrt{2/\pi}$ for standard normal.