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!

  • $\begingroup$ 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. $\endgroup$
    – Kurt G.
    Feb 16 at 20:29


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