Suppose we have an OU process given by the stochastic differential equation
$dr_t = \kappa(\theta-r_t)dt + \sigma dW_t$.
I think that the distribution of $D(t,T) := \int_t^T r_s\;ds$ is normal (I still want to prove that. I thought maybe this can be done by seeing the integral as a Riemann sum, which is clearly normally distributed since it is a finite sum of normals, and then showing that the limit is still normal. Or is this maybe a standard result?).
Regardless of the distribution, I need the mean and variance of $D(t,T)$ given the information known at time $t$. I already calculated the mean by using stochastic Fubini and the solution to the SDE:
$r_t = r_0 e^{-\kappa t} + \theta (1-e^{-\kappa t})+\sigma \int_0^t e^{-\kappa (t-s)}dW_s$.
So I got \begin{align*} E_t D(t,T) = E_t\left[\int_t^T r_s\;ds\right]&=\int_t^T E_t[r_s]\;ds\\ &=\int_t^T r_t e^{-\kappa(s-t)}+\theta(1-e^{-\kappa(s-t)})\;ds\\ &=-\frac{r_t}{\kappa}(e^{-\kappa (T-t)}-1) + \theta\left(T-t+\Big(\frac{1}{\kappa}e^{-\kappa (T-t)}-\frac{1}{\kappa}\Big)\right)\\ &=-(\theta-r_t)\frac{1}{\kappa}(1-e^{-\kappa(T-t)}) + \theta(T-t). \end{align*}
I am stuck with the calculation of the variance. Even if I knew it really is normal, I can't seem to exploit the fact that we are dealing with a $\chi^2$ random variable.
Plug in the expression for $r_s$ into the integral first, and then switch integrals (again by stochastic Fubini). Then applying Ito's isometry and doing some elementary calculus should do the trick.