I have been learning basic stochastic analysis, and we have only been taught about Ito formula. The professor told us how can we solve this question below using it, but I miss it. Can anyone help me?

$$ dr_t = (\alpha - \beta r_t) dt + \sigma dW_t,$$

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  • $\begingroup$ Possible duplicate: math.stackexchange.com/questions/1148294/… $\endgroup$ – Brenton Jun 21 '16 at 1:39
  • $\begingroup$ Hint. It is good to think about the deterministic counterpart $dy_t = (\alpha - \beta y_t) dt$ first. Solutions of this equation are of the form $y_t = \frac{1}{\beta}(\alpha - A e^{-\beta t})$. This means that $r_t$ is a stochastically perturbed version of this solution, and we can hope that $u_t$ defined by $r_t = \frac{1}{\beta}(\alpha - A e^{-\beta t}) u_t$ would simplify the problem. This is indeed the case, as in the answer above. $\endgroup$ – Sangchul Lee Jun 21 '16 at 1:41

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