Mean reverting Ornstein-Uhlenbeck SDE

Question

A mean reverting Ornstein-Uhlenbeck SDE is given by $$=(−)+;_0=,$$ where m and are positive constants and is a standard Brownian motion in 1 dimension. I have obtained the solution of this equation, $$X_t = xe^{-t} + m(1-e^{-t}) + \sigma \int_0^t e^{-(t-s)} dW_s.$$ My questions are: Is my solution correct? What is the long time behaviour of the solution? How to obtain the equation of the second moment?

Answer 1

Hint

I didn't check into detail the solution you found, but at least, the solution is something of this form. Now, one can prove that if $f\in L^2[0,t]$, then $$\int_0^tf(s)\,\mathrm d W_s\sim \mathcal N\left(0,\int_0^t f(s)^2\,\mathrm d s\right).$$