I have this process:
$dx_t = -\frac{k}{2}x_tdt + \frac{\beta}{2}dz_t$
and must prove it's normally distributed with first two moments:
$\mu = e^{-\frac{1}{2}kt}x_0$
$\sigma^2 = \frac{\beta^2}{4k}(1-e^{-kt})$
I tried to multiply $x_t$ by $e^{kt}$ and apply Ito's Lemma to this 'product process' in order to eventually recover back $x_t$ by taking exponentials.
The normality is straightfoward; the variance is ok but the mean isn't since I'm left with an integral whose integrand includes $x_t$ and I'm stuck.
I don't know whether I made some mistakes or adopted the wrong approach since the beginning.