Suppose we have the solution to the ordinary Vasicek model:
$$r_t = r_0 e^{-a t} + b(1 - e^{-a t}) + \sigma \int^{t}_0 e^{-a (t-s) } dW_s$$
How do I use the Ito's lemma to recover the SDE
$$dr_t = a(b - r_t)dt + \sigma dW_t$$
Thank you for your help.
Multiply with $e^{at}$ and take the derivative, then apply simple product rules with one deterministic factor and further algebraic manipulations: $$ d(e^{at}r_t)=b\,d(e^{at}-1)+σ\,e^{at}\,dW_t\\ e^{at}(dr_t+ar_t\,dt)=ab\,e^{at}\,dt+σ\,e^{at}\,dW_t\\ dr_t=a(b-r_t)\,dt+σ\,dW_t $$
