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Hi all,

I think that the definition of fractional Brownian Motion is widely known (for example as a Gaussian Process with particular variance covariance stucture parametrized by the so-called Hurst index).

Heuristically, you can think of those processes as Gaussian processes with long (or short) memory depending on the value of their Hurst Index, and for Hurst index equal to 1/2 you get classical Brownian Motion (which has no memory).

I was wondering what would be the definition for "fractional Poisson Processes" and what stylised facts about fractional Brownian Motion one should consider in extending the definition to Poisson process.

If any reference exists about this, this is just fine for me.

I have no other motivation than curiosity on this topic.

Regards

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A google search for fractional Poisson process gives a lot of results. Fractional Brownian motion is weird enough, with stochastic integrals no longer being martingales; yuck. –  Alex R. Jan 25 '11 at 18:49
    
True but google doesn't tell us what are the best papers or the "level" associated with them ... Moreover an answer which indicates the canonical properties of the fBM that should be extended in order to get a fractional Poisson process would give a great insight before reading google's results on fPP Best Regards –  The Bridge Jan 26 '11 at 7:25
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2 Answers

Hello Everyone,

To find the definition of Fractional Poisson Process and its first two moments as well as the Compound Fractional Poisson Process go to http://pi.314159.ru/laskin3.pdf To find more on the topic you may Google with key words: fractional Poisson distribution. I hope it helps.

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Here is a thesis containing (in Section 2) an overview of different definitions of fPP.

My personal favorite is the "Standard Fractional generalization I" defined in 2.2. The reason is that there seems to be (I failed to find any relevant results) an isomorphism between this version and fBm similar to the (usual) Wiener-Poisson isomorphism.

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