Let us consider a process $\{X(t);t\in[0,1]\}$. Let us define $X(t)=\int_0^tg(u)dB(u)$ where $g$ is a deterministic function and $B$ is Brownian bridge process.
Then what will be the mean and variance function of $X(t)$?
If we assume $B$ to be a Brownian motion the mean function is zero and variance function (from Eto's isometry), we can get $$V(X(t))=\int_0^tg^2(u)du.$$
Stuck in the case of Brownian bridge.