Let a stochastic interest rate process $R(t)$ satisfies $dR(t) = (\alpha - \beta R(t))dt + \sigma dW(t)$.

Question

$\alpha$, $\beta$, and $\sigma$ are positive constants.

How do I compute $d(e^{\beta t} R(t))$ and simplify the result such that I have a formula for $d(e^{\beta t} R(t))$ not including $R(t)$?

Answer 1

$$\begin{align}d(e^{\beta t} R(t)) & = \beta e^{\beta t} R(t)dt + e^{\beta t} dR(t) \\ & = \beta e^{\beta t} R(t)dt + e^{\beta t} dR(t) \\ & = e^{\beta t} (\alpha dt + \sigma dW(t)-dR(t)) + e^{\beta t} dR(t) \\ & = \alpha e^{\beta t} dt + \sigma e^{\beta t} dW(t) \end{align}$$