How to show the expected value of a hitting time Brownian motion?

Question

We have $W_t$ as a Brownian motion and $$T_{−a,b} = \inf \{t ≥ 0 : W_t \not\in [−a, b]\}\qquad a, b > 0$$ How do you show $\mathbb{E} (W_{T_{-a,b}}) = 0$?

Answer 1

Hint: Deduce from the optional stopping theorem that$$\mathbb{E}(W_{T_{-a,b} \wedge R})=0$$ for any $R>0$. Apply the dominated convergence theorem in order to conclude $$\mathbb{E}(W_{T_{-a,b}})=0.$$