# 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

## 2 Answers

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.

• This is very helpful, thank you. Can you explain why it is not possible to estimate the autocorrelation when the mean is changing? May 17 '18 at 13:47
• The assertion "The autocorrelation is only defined for constant mean processes." is incorrect, here on dsp.SE and on stats.SE as well as on math.SE. May 17 '18 at 15:39
• Hi: It's due to the definition of autocorrelation See first page of link below.. According to this document's first definition, it can be defined, but that's a very odd definition of autocorrelation ? that def is puzzling to me and I would ignore it. I know of no process where the mean is changing at each point in time but you know what it is ? . It's the second definition is I was referrring to. that is why it can't be defined. I hope that the second definition helps to understand what I was said earlier. Here's link: gauss.stat.su.se/gu/e/slides/Lectures%208-13/… May 17 '18 at 18:14
• Hi Dilipe: Yes, you are correct. I see it's defined on the link I sent. But, as far as I know, no one calculates autcorrelation when the mean is changing at different points in time. If he wants to do that, he can ( OP. see first def in link ), but I don't know what he's going to use for the mean. can you explain in more detail the application of autocorrelation in the case where the mean is changing ? Atleast in time series apps, there are none. May 17 '18 at 18:17
• Hi Dilipe: You are correct squared. I The link ( below ) provides the expression on page 12. .To the OP: You can think of the formula as "correcting" for the fact that mean is changing over time. I didn't look at the derivation but that's the idea. maths.ox.ac.uk/system/files/attachments/Thierfelder_v1.08.pdf. Thanks to Dilipe for correcting me. You live and learn. In statistical time series, one would de-trend ( atleast as far as my experience goes ) but in other fields, this is not the case and mathematical finance is one of them. My apologies for confusion and thanks to Dilipe. May 17 '18 at 18:29

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.