If given the Vasicek Interest rate model $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ how do I use Ito's lemma to find $d(e^{\beta t}R(t))$? If given the Vasicek Interest rate model $dR(t)=(\alpha-\beta R(t))dt +\sigma dW(t)$ how do I use Ito's lemma to find $d(e^{\beta*t}R(t))$ and simplify so it is a solution that does not include R(t).
To further this question, how would I go about integrating the step above to solve for R(t) then use this to find the expectation and variance of R(t) along with it's limiting behaviors
Could someone please walk me through the steps of doing this! It would be very much appreciated
EDIT: I think I figured out how to do the first part:
let $f(R(t))=e^{\beta t}R(t)$
$f'(R(t))=e^{\beta t}$
$f''(R(t))=0$
$f'(t)=\beta e^{\beta t}R(t)$
then, $d(e^{\beta t} R(t)) = \beta e^{\beta t} R(t)dt + e^{\beta t} dR(t) + 0$
$=e^{\beta t} dR(t) + \beta e^{\beta t} R(t)dt$
Now i sub dR(t) into the above equation and simplify to get
$d(e^{\beta t} R(t)) = e^{\beta t}((\alpha - \beta R(t))dt + \sigma dW(t)) + \beta e^{\beta t} R(t)dt = \alpha e^{\beta t}dt + \sigma e^{\beta t}dW(t)$
Is the above correct notation so far?
 A: Let $Y_t = e^{\beta t}R_t$, hence
$$dY_t = \alpha e^{\beta t} dt + \sigma e^{\beta t} dW_t,$$
$$Y_t = Y_0 +  \alpha\int\limits_0^t e^{\beta s} ds + \sigma \int\limits_0^te^{\beta s} dW_s,$$
$$e^{\beta t}R_t = R_0 +  \alpha\int\limits_0^t e^{\beta s} ds + \sigma \int\limits_0^te^{\beta s} dW_s,$$
$$e^{\beta t}R_t = R_0 +  \frac{\alpha}{\beta} e^{\beta t} + \sigma \int\limits_0^te^{\beta s} dW_s,$$
$$R_t = e^{-\beta t}R_0 +  \frac{\alpha}{\beta} + \sigma \int\limits_0^te^{\beta (s-t)} dW_s.$$
This is the final expression for $R_t$. 
$$\mathbb{E}R_t = e^{-\beta t}\mathbb{E}[R_0] +  \frac{\alpha}{\beta},$$
therefore $\mathbb{E}R_t \rightarrow \frac{\alpha}{\beta}.$ 
A: Vasicek Interest rate model has the desirable property that the interest rate is mean-reverting. When $R(t) = \frac{\alpha}{\beta} $ the drift term in $dR(t)$ is zero. When $R(t) > \frac{\alpha}{\beta} $ this term is negative, Which pushes $R(t)$ back toward $\frac{\alpha}{\beta} $. Likewise when $R(t) < \frac{\alpha}{\beta} $ this term is positive, which again pushes $R(t)$ back toward $\frac{\alpha}{\beta} $
