Finding the limit $\lim_{t\to ∞} \mathbb{E}[R_t]$ of an SDE

Question

I have the SDE

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

In this equation, $R_0 = r$ in which $r > 0$

Can someone please help me find the $\lim_{t\to ∞} \mathbb{E}[R_t]$?

Thus far I have calculated the expectation:

From Itô's lemma $$d(e^{\beta t}R_t)=\beta e^{\beta t}R_tdt+e^{\beta t}dR_t+\underbrace{d[{{e}^{\beta t}},{{R}_{t}}]}_{0}$$ $$d(e^{\beta t}R_t)=e^{\beta t}dt+\sigma e^{\beta t} dB_t$$ Then $$e^{\beta t}R_t=R_0+\frac{1}{\beta}(e^{\beta t}-1)+\sigma \int_{0}^{t}e^{\beta s}dB_s$$

$$R_t=R_0e^{-\beta t}+\frac{1}{\beta}(1-e^{-\beta t})+\sigma \int_{0}^{t}e^{-\beta (t-s)}dB_s$$

$$\mathbb{E^Q}[R_t]=re^{-\beta t}+\frac{1}{\beta}(1-e^{-\beta t})$$

What do I do from here?

Answer 1

Assuming that $\beta>0$, then \begin{align*} \lim_{t\rightarrow \infty} E(R_t) &= \lim_{t\rightarrow \infty} \left[R_0 e^{-\beta t} + \frac{1}{\beta} (1-e^{-\beta t}) \right]\\ &= \frac{1}{\beta}. \end{align*}