I'm working on the problem for a university project, I struggling for several days, I would post the same question with different coefficients from the problem I currently working on, so please show me the process to solve it and then I would learn by myself, your help is really important to me, please!
The questions are below:
Suppose $(W_t)$ is a standard Brownian motion, $t > 0$. Let $H$ be the first time the process $$X_t = W_t^2 + 2t$$ hits the value $a > 0$, what is $\mathbb E[H]$?
I tried to do it using the optional stopping theorem, but I got stuck, please help me if you know how to do it! Thanks so much!