Not using stochasstic integral how to prove $E\int_0^T W^2(t)dt<+\infty$?

Question

Can anyone help me to prove this?

Suppose $W_t$ ~ $N(0,t)$, then not using stochasstic integral (or anything related with Ito) how to prove $E\int_0^T W^2(t)dt<+\infty$?

Thanks.

Answer 1

By Fubini's Theorem (the integrand is positive): \begin{align} E\int_0^TW^2(t)dt=\int_0^T(EW^2(t))dt=\int_0^Ttdt=T^2/2<\infty. \end{align}