Let [itex]\phi(t)[/itex] be a Brownian Walk (Wiener Process), where [itex]\phi\in[0,2\pi)[/itex]. As such we work with the variable [itex]z(t)=e^{i\phi(t)}[/itex]. I would like to calculate(adsbygoogle = window.adsbygoogle || []).push({});

[itex]E(z(t)z(t+\tau))[/itex]

This is equal to [itex]E(e^{i\phi(t)+i\phi(t+\tau)})[/itex] and I know that

[itex]E(e^{i\phi(t)})=e^{-\frac{1}{2}\sigma^{2}(t)}[/itex], where the mean is 0 and [itex]\sigma^{2}(t)=2Dt[/itex].

However, I have been stuck a week on how to proceed, any thoughts?

Thank you :)

Aim For Clarity

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# Autocorrelation of a wiener process

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