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Im not sure its very Paul clear. By your definition, a Brownian Motion is stationary. In fact, for a stochastic process, stationarity is defined as statistically invariant under translations. Try calculating this for the Brownian Motion and OU Process: $\forall A \in \mathbb{R}^N$ $Pr\{X_1, ..., X_n \in A\} = Pr\{X_{1+h}, ..., X_{n+h} \in A\}$ If those ...


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I think you misunderstood the definition. Be stationary does not mean not depend of the time as you can check here. (Sorry for putting an wikipedia link here as I suppose you may have read it) Another way to think is that the law any increment of the process is given by a same function of the difference of time. More precisely $\forall ~t_2\geq t_1,$ : ...



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