Suppose the process $x_t=u_t+v_t\int\limits_{0}^tg_t\,dW_t$ is given. Here $W_t$ is the standard Wiener process, $u,v,g$ are some deterministic funcions, and $g$ is such that the integral is well defined. What I am interested in is the integral $I_t=\int\limits_0^tx_sds$, or, more precisely, it's distributional properties. I see it is the normal random variable, since $x_t$ is a gaussian process. What I need is $Var\bigl[I_t\bigr]$. Is there an elegant way to derive it in a closed form?
I see, that $I_t$ is the limit of $\sum\limits_{k=0}^{n-1}x_{k}\Delta s_k$, and thus $Var[I_t]$ is the limit of $\sum\limits_{k=0}^{n-1}Var[x_k](\Delta s_k)^2+2\sum\limits_{k<l}\Delta s_k\Delta s_l\, \mathrm{cov}(x_k,x_l)$, where both $Var[x_k]$ and $\mathrm{cov}(x_k,x_l)$ can be derived in a closed form, but this seems to be a bulky derivation.