Covariance of a random function

Question

Suppose $X(s)=\int_0^1 G(s,t)\, dW(t)$, where $W(t)$ is Brownian motion, then what is the variance of $X(s)$ and the covariance of $X(s)$ and $X(r)$.

Note that this is not the usual Ito integral question, since integration is over a fixed integral. $X$ is a random function obtained by smoothing a realization of Brownian motion over $[0,1]$

Answer 1

If $G$ is deterministic, this is very much "the usual Ito integral question", actually the very definition shows that, for every $s$ and $r$, $$ E(X(s)X(r))=\int_0^1G(s,t)\,G(r,t)\,\mathrm dt. $$