Let us consider the following SDE: $$ d x(t)=-\alpha \cdot(x(t)-\mu) \cdot d t+\sqrt{2 \cdot \alpha} \cdot \sigma \cdot d W(t) $$ Its solution is a stationary stochastic process called the Ornstein-Uhlenbeck process. How could I calculate the autocorrelation of the Ornstein Uhlenbeck process?
My attempt. Thanks to @Chaos comment, I deleted the previous wrong attempt and I tried to solve the equation following analytic solution to Ornstein-Uhlenbeck SDE. In doing so, I obtained the equation: $$x(t)=\mu+e^{\alpha(s-t)}\left(x(s)-\mu\right)+\sqrt{2\alpha}\sigma e^{-\alpha t}\int_s^t e^{\alpha u}dW(u)$$ The autocorrelation is: $$\mathrm{R}_{x}(\tau)=\mathrm{E}\left[\mathrm{x}(t) \mathrm{x}^{*}(t+\tau)\right]$$ But... how can I proceed?