Prove that the expectation of Cox Ingersoll Ross interest rate model is $r_0e^{-at} + b(1-e^{-at})$

Question

The Cox Ingersoll Ross is given in the form:

$$dr_t = a(b-r_t)dt + \sigma \sqrt{r_t}dW_t$$

According to Wikipedia, the expected value of it i.e $E[r_t|r_0]$ is

$$r_0e^{-at} + b(1-e^{-at})$$

but I don't really understand how it's derived. Can you give some hints?

Answer 1

Hint

Remark that

$$\mathbb E[r_t]=\mathbb E[r_0]-a \int_0^t\mathbb E[r_s]\,\mathrm ds+abt$$

and use Gronwall inequality.