# How to show that Ornstein-Uhlenbeck is not a martingale with $E[X(t)]=E[X(s)]$ for all $s,t\in [0,\infty)$?

If we are given that $$X=\{X(t)\}_{t\in[0,\infty)}$$ is an O-U-Process, which is defined as the continuous centered Gaussian process with covariance function given by

$$\Gamma(s,t)=\frac{\sigma^2}{2\kappa}e^{-k(t+s)}(e^{2\kappa(t\wedge s)}-1),$$ where $$\sigma,\kappa>0$$, how does one show that this is not a martingale with $$E[X(t)]=E[X(s)]$$ for all $$s,t\in [0,\infty)$$?

Since it is centered, we immeadiately have $$E[X(t)]=E[X(s)]=0$$, for all $$s,t\in[0,\infty)$$ and I could show that it is not a martingale, if the process is explicitly given as for example $$X(t)=e^{-\kappa t}W(\frac{\sigma^2}{2\kappa}(e^{2\kappa t}-1)),$$ where $$W$$ is a Brownian motion but without it I have no clue. So the question is, only knowing the covariance function, how does one show this? Any help is appreciated!

• There are two flavors of O.U. process. See this detailed answer. None of those two is a martingale, despite both of them being centered. Reason: the drift. Feb 16 at 20:29