Is there a general (integral) solution to $du(t)= -a(t)u(t)\,dt +\sigma(t)\,dz$? Is the following $u(t)=e^{-\int\_{t\_0}^{t} \alpha(s) \, ds}u(t\_0)+\int\_{t\_0}^t \sigma(v) e^{-\int\_v^t a(s) \, ds} \, dz(v)$ correct (which I have seen claimed without a justification)? z(t) is the standard Wiener process. Is there a good reference for it? 



A solution to stochastic PDE $du(t)= a(t)u(t)\,dt +s(t)\,dz$
About the differentiation formula of $u(t)$: Since 
\begin{align}
u(t)&=e^{-\int\_{t\_0}^ta(s)\mathrm{d}s}u(t\_0)+\int\_{t\_0}^t\sigma(v)e^{-\int\_{v}^ta(s)\mathrm{d}s}\mathrm{d}Z(v)\\
&=e^{-\int\_{t\_0}^ta(s)\mathrm{d}s}\Bigl[u(t\_0)+\int\_{t\_0}^t\sigma(v)e^{\int\_{t\_0}^va(s)\mathrm{d}s}\mathrm{d}Z(v)\Big]\stackrel{\text{def}}{=}x(t)y(t).\\
\mathrm{d}x(t)&=-a(t)e^{-\int\_{t\_0}^ta(s)\mathrm{d}s}\mathrm{d}t=-a(t)x(t)\mathrm{d}t,\\
\mathrm{d}y(t)&=\sigma(t)e^{\int\_{t\_0}^ta(s)\mathrm{d}s}\mathrm{d}Z(t)=\sigma(t)[x(t)]^{-1}\mathrm{d}Z(t).
\end{align}
Therefore,
$$
du(t)=y(t)\mathrm{d}x(t)+x(t)\mathrm{d}y(t)
=-a(t)u(t)\mathrm{d}t+\sigma(t)\mathrm{d}Z(t).
$$


