Expectation of indicator of the brownian motion inside an interval

Question

Suppose $W_t$ is your usual brownian motion and that you have the following process:

$$ \theta_t = \int_0^t \mathbb{1}_{\alpha \leq W_s \leq \beta} ds $$

How can I calculate the expectation $E[\theta_T]$?

Answer 1

You may apply the Fubini theorem. Its hypotheses are satisfied. So you get:

$$E[\theta_T] = \int_0^T P[W_s \in [\alpha,\beta]] ds = \int_0^T \int_\alpha^\beta \frac{1}{\sqrt{2\pi s}} \exp(-\frac{x^2}{2s}) ~dx ~ds. $$

Etc.