Let $W$ be a Brownian Motion and $a,b \in \mathbb{R}$. Solve the following SDE:
$\text{d}X_t = -aX_t\text{dt} + b\text{d}W_t$.
I am familiar with the it$\hat{o}$ lemma, but I can never figure out how to choose a function for $f(t,x)$. Could someone walk me through the steps of this simple SDE, I would really appreciate it :)!
Update: I have solved it in the following way, although I am still a little unsure as to whether this solution is correct. Ok so this is my working:
$dXt+aXtdt=bdWt$
I then used the integrating factor with p(t)=at. Multiplying both sides by this I got:
$e^{at}dXt+aXte^{at}=be^{at}dWt$,
which I then noticed is equal to the product rule on the LHS:
$d(e^{at}Xt)=be^{at}dWt$
and then by integrating I obtained:
$e^{at}Xt=X0+\int^{t}_{0}be^{as}dWs$.
So my final solution is: $Xt=X_0e^{-at}+e^{-at}\int^{t}_{0}be^{as}dWS$. Is this correct?