I am trying to prove a statement and a term comes up for which I need to show that it is finite.
Let $(W_s)_{s\geq0}$ be a $\mathbb{P}$-Brownian motion. Is for any $t\geq0$ the following integral finite
$$\mathbb{E}\left[\int_0^t (e^{W_s})^2 ds\right].$$
I know that using It's Isometry one can conclude
$$\mathbb{E}\left[\int_0^t (e^{W_s})^2 ds\right]=\mathbb{E}\left[(\int_0^t e^{W_s} dW_s)^2\right].$$
But I don't see how this helps me.
This is an ordinary Lebesgue/Riemann integral (of a non-negative continuous function w.r.t. Lebesgue measure). We can apply Fubini's Theorem to evaluate it. $\mathbb E\int_0^{t} (e^{W_s})^{2}ds=\int_0^{t}\mathbb E e^{2W_s} ds=\int_0^{t}e^{2s}ds=\frac {e^{2t}-1} 2$.