I have the stochastic process given by
$$ \dot{x} = ax + v$$
where $v$ is a zero-mean, Gaussian white noise signal. What is the autocorrelation of $x$?
I believe the process is to first find the solution of the ODE, which is
$$ x(t) = e^{at} x(0) + \int_0^t e^{a(t-\tau)} v(\tau)\, d\tau $$
Then, the autocorrelation function is
$$R_{xx} = \int x(t+\tau)x(t)\, dt$$
However, I think I need to use the statistical definition since I have a stochastic process
$$R_{xx} = E[x(t+\tau)x(t)]$$
But, I am not sure how to solve this.
