How to deduce the expectation of a stochastic equation

Question

I am having a difficult time deducing the expectation, $\mathbb{E}[R_t]$, of the following stochastic equation:

$$dR_t = (1 - \beta R_t)dt + \sigma dB_t$$

$R_0 = r$, with $r > 0$.

Please help me with this!

Answer 1

By application of Ito lemma we have $$d(e^{\beta t}R_t)=\beta e^{\beta t}R_tdt+e^{\beta t}dR_t+\underbrace{d[{{e}^{\beta t}},{{R}_{t}}]}_{0}$$ $$d(e^{\beta t}R_t)=e^{\beta t}dt+\sigma e^{\beta t} dB_t$$ therefore $$e^{\beta t}R_t=R_0+\frac{1}{\beta}(e^{\beta t}-1)+\sigma \int_{0}^{t}e^{\beta s}dB_s$$ In other words $$R_t=R_0e^{-\beta t}+\frac{1}{\beta}(1-e^{-\beta t})+\sigma \int_{0}^{t}e^{-\beta (t-s)}dB_s$$ thus $$\mathbb{E^Q}[R_t]=re^{-\beta t}+\frac{1}{\beta}(1-e^{-\beta t})$$