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The fractional Poisson process (FPP) is a counting process with independent and identically distributed inter-event times following the Mittag-Leffler distribution. This process is very useful in several fields of applied and theoretical physics including models for anomalous diffusion. Contrary to the well-known Poisson process, the fractional Poisson process does not have stationary and independent increments. It is not a Levy process and it is not a Markov process. In this letter, we present formulae for its finite-dimensional distribution functions, fully characterizing the process. These exact analytical results are compared to Monte Carlo simulations.
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We introduce a non-homogeneous fractional Poisson process by replacing the time variable in the fractional Poisson process of renewal type with an appropriate function of time. We characterize the resulting process by deriving its non-local governing equation. We further compute the first and second moments of the process. Eventually, we derive the distribution of arrival times. Constant reference is made to previous known results in the homogeneous case and to how they can be derived from the specialization of the non-homogeneous process.
The fractional non-homogeneous Poisson process was introduced by a time-change of the non-homogeneous Poisson process with the inverse $alpha$-stable subordinator. We propose a similar definition for the (non-homogeneous) fractional compound Poisson process. We give both finite-dimensional and functional limit theorems for the fractional non-homogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombes theorem. Eventually, some of the limit results are verified in a Monte Carlo simulation.
This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing Levy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.
In this paper, we quantitatively investigate the properties of a statistical ensemble of stock prices. We focus attention on the relative price defined as $ X(t) = S(t)/S(0) $, where $ S(0) $ is the initial price. We selected approximately 3200 stocks traded on the Japanese Stock Exchange and formed a statistical ensemble of daily relative prices for each trading day in the 3-year period from January 4, 1999 to December 28, 2001, corresponding to the period in which the {it internet Bubble} formed and {it crashes} in the Japanese stock market. We found that the upper tail of the complementary cumulative distribution function of the ensemble of the relative prices in the high value of the price is well described by a power-law distribution, $ P(S>x) sim x^{-alpha} $, with an exponent that moves over time. Furthermore, we found that as the power-law exponents $ alpha $ approached {it two}, the bubble burst. It is reasonable to assume that when the power-law exponents approached {it two}, it indicates the bubble is about to burst. PACS: 89.65.Gh; Keywords: Market crashes, Power law, Precursor
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