 Solution

The Skorokhod embedding theorem states that if $\mu$ is a centered probability law on $\mathbb{R}$ with finite second moment, there is a Brownian stopping time $T$ such that $B_T\sim \mu$, the stopped process $(B_{t\wedge T})$ is uniformly integrable, and $\mathbb{E}T=\int x^2\mu (dx)$.