thank you very much for clicking on my question. I'm working on this paper (https://www.duo.uio.no/bitstream/handle/10852/10566/pm12-05.pdf?sequence=1) (Page 3) and want to solve the following SDE:
$dT(t)=db(t)-a(t)(T(t)-b(t))dt+\sigma(t)dB(t)$
The solotion is: $T(t)=b(t)+(T(0)-b(0))e^{-\int_{0}^{t} a(s)ds}+e^{-\int_{0}^{t} a(s)ds} \int_{0}^{t} \sigma(u)e^{\int_{0}^{u} a(s)ds} dB(u)$
In my case a(t) and b(t) are deterministic functions and B(t) the standard brownian motion. It is described that the SDE is solved using the Ito formula.
My question is why I can use the ito formula at all since the SDE has to represent the following form: $dT(t)=a(t,T(t))dt+b(t,T())dB(t)$? Or how do I have to deal with $db(t)$?
The second question is how to solve the SDE with Ito? with $Y(t)=T(t)e^{\int_{0}^{u} a(s)ds}$?
As an exercise, I solved the ornstein-Uhlenbeck SDE with ito:
$dX_t=\theta (\mu-X_t)dt+ \sigma dW_t$ with $Y_t=X_t e^{\theta t}$
Thank you very much