What would be a fractional Poisson Process like

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.

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

A standard Poisson process is a renewal process with exponential distributed waiting times. Fractional Poisson process (FPP) is also a renewal process with Mittag-Leffler waiting times. Note that Mittag-Leffler distribution is a heavy tailed generalization of exponential distribution. Further, let N(t) be a standard Poisson process and $E_{\alpha}(t) = \inf\{s \geq 0: S_{\alpha}(s)>t\}$ be the first-exit time of a stable subordinator $S_{\alpha}(t)$ then the time-changed process $N^*(t) = N(E_{\alpha}(t))$ is also a characterization of FPP.

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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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The link doesn't work. Could you please tell me author and title of the thesis? – user74045 Mar 24 at 22:48
@user74045, the name is Dexter Cahoy. I've updated the link. – zhoraster Mar 25 at 4:35

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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The inverse $\alpha$-stable subordinator approach as a possible definition of the fractional Poisson process (fPP) is (arguably) the most commonly known and all answers so far seem to cover this. However, since your question seems to be motivated by the notion of fractional Brownian motion (fBM), it is worth mentioning that there is a completely different definition of fPP:

The suggested fPP in (Wang, Wen, and Zhang 2006) shares more properties with the fBM like covariance structure, self-similarity and stationary increments. Also it is shown that fPP converges to fBM in distribution.

Moreover, the inverse subordinator method can also be applied to Brownian motion (i.e. consider $(B(E_\alpha(t))$, where $(E_\alpha(t))$ is the inverse $\alpha$-stable subordinator) which gives us something that goes under the name of time-fractional diffusion rather than fractional Brownian motion (see (Meerschaert and Straka 2013) and related work). This stochastic process is self-similar, but does no longer have stationary increments as $(E_\alpha(t))$ has not.

To conclude, as the literature for both definitions acknowledges one another, it seems that both are worth considering.

References:

• Meerschaert, M. M. and P. Straka (2013). Inverse stable subordinators. Math. Model. Nat. Phenom. 8 (2), 1–16.
• Wang, X.-T., Z.-X. Wen, and S.-Y. Zhang (2006). Fractional Poisson process. II. Chaos Solitons Fractals 28 (1), 143–147.
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