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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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For both time-series, just plot the log returns. You will see that one is not a Random-Walk .. the S&P500 since you will get values that far beyond the normal distribution. Just watch this video by Benoit Mandelbrot (starting at 11min:54sec). Looking at both graphs, your eyes can fool you making you believe that both are generated by Random Walks...

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I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is perfectly symmetric whereas the real one takes longer for going up and then crashes in a relatively shorter time frame. This is a known phenomenon in real financial ...

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You have just luckily created 1 path of the random walk by chance that fitted the S&P. You can create another random walk and it will look much different. The efficient markets hypothesis predicts that stock prices behave as random walks, so it is likely that S&P looks similar to that. However, one cannot predict the next step to make a profit, ...

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