Expected value and variance of a stopping time

Question

Let $B = \{B(t) : t \ge 0\}$ be a Brownian motion and $\sigma_a$ be a the stopping time such that $\sigma_a = \inf\{t \ge 0: B(t) \le t - a\}$ for $a>0$. I want to find $\mathbb{E}[\sigma_a]$ and $\text{Var}[\sigma_a]$ but I really have no idea where to start. Could one of you give me a hint?

Answer 1

Since $B$ is a martingale, you can apply Doob's optional stopping theorem to get: $$E B_{\sigma_a} = E B_0 = 0$$ Since $B_{\sigma_a} = \sigma_a - a$ almost surely, that gives you an equation to solve for $E \sigma_a$. Applying the same technique to the martingale $B_t^2-t$ gives you an equation for the second moment.

This leaves out a few important details. Most versions of the OST won't apply directly, but will need a sequence stopping times (e.g., $\sigma^M_a=\{t\geq0 : B(t)\not\in(t-a,M)\}$), and you'll need to prove both something about $\sigma_a$ (e.g., that it's almost surely finite) and apply the dominated convergence theorem. See the outline here for a detailed outline for a similar problem.