I am wondering how we can solve the SDE:
$$ dX_t = -\beta(X_t-\alpha)dt + \sigma dW_t $$
where $W_t$ is a Wiener process, and $\beta >0$, $\alpha \in \mathbb{R}, \sigma>0$ are constants. It seems that we can define it according to the OU process, but with a substitution. Is a substitution doable?
HINT :
Introduce the process $Y_t=e^{\beta t}X_t$. When you apply Ito's lemma , you have
$$dY_t=\beta Y_tdt+e^{\beta t}dX_t=\beta Y_tdt+ e^{\beta t}(-\beta(X_t-\alpha)dt + \sigma dW_t)$$
Thus, $$dY_t=\alpha e^{\beta t}+e^{\beta t}\sigma dW_t$$
$$Y_t=Y_0+\int_{0}^{t}\alpha e^{\beta u}du+\int_{0}^{t}{\sigma e^{\beta u}dW_u}$$
Finally, you rewrite it in terms of $X_t$
$$X_t=X_0e^{-\beta t} +\int_{0}^{t}\alpha e^{-\beta(t- u)}du+\int_{0}^{t}{\sigma e^{-\beta(t-u)}dW_u}$$