# Ornstein Uhlenbeck with drift

The Ornstein-Uhlenbeck (OU) process

$dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t$

generates coloured noise with autocorrelation function $R(t) = \langle X_t,X_{t'}\rangle = \sigma^2 e^{-|t-t'|/\mu}$.

I would be interested in modelling a process $Y_t = \alpha t + X_t$. After some searching, I came across the terms "OU process with drift" or "trending OU process".

Does anyone know what is the autocorrelation function of the process $Y_t$? Thank you!

• my recollection of OU is that it is simply white noise filtered with a 1st-order, 1-pole (analog) filter. similarly how Brownian motion is white noise filtered with an (analog) integrator. May 17 '18 at 0:38

Hi: This is a statistics viewpoint and not a DSP one. I just say that because there are termimology issues that have caused confusion with my answers in the past. The autocorrelation is only defined for constant mean processes. A process with drift doesn't have a constant mean so you would need to detrend the process first which would bring you back to the zero mean case. One way to do that would be run a regression with time as your "x", in order to estimate $\alpha$. Then, once you have the estimate, you can subtract the trend off. I'm not sure if this is the answer you wanted to hear but the trend makes it not possible to estimate correlations because they can't be calculated because the mean of the process is not constant.
Ignoring the Ornstein-Ulhenbeck connection, the process $\{Y(t) = \alpha t X(t)\}$, where $\{X(t)\}$ is a zero-mean wide-sense-stationary (WSS) process with autocorrelation function $R_X(\tau) = E[X(t)X(t+\tau)]$ has autocorrelation function \begin{align} \require{cancel}R_Y(t_1, t_2) &= E[Y(t_1)Y(t_2)]\\ &= E\big[(\alpha t_1 + X(t_1))(\alpha t_2 + X(t_2))\big]\\ &= \alpha^2 t_1t_2 + \alpha t_1\cancel{E[X(t_2)]} + \alpha t_2\cancel{E[X(t_1)]} + E[X(t_1)X(t_2)]\\ &= \alpha^2 t_1t_2 + R_X(t_2-t_1) \end{align} showing that $\{Y(t)\}$ is not a WSS process. More simply, of course, $E[Y(t)]$ is not a constant and so $\{Y(t)\}$ also fails to meet that requirement for WSS processes.