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Simulation and estimation for the fractional Yule process

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      Abstract

      In this paper, we propose some representations of a generalized linear birth process called fractional Yule process (fYp). We also derive the probability distributions of the random birth and sojourn times. The inter-birth time distribution and the representations then yield algorithms on how to simulate sample paths of the fYp. We also attempt to estimate the model parameters in order for the fYp to be usable in practice. The estimation procedure is then tested using simulated data as well. We also illustrate some major characteristics of fYp which will be helpful for real applications.

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      Fractional Poisson processes and related planar random motions

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        Fractional diffusion equations and processes with randomly varying time

        In this paper the solutions \(u_{\nu}=u_{\nu}(x,t)\) to fractional diffusion equations of order \(0<\nu \leq 2\) are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order \(\nu =\frac{1}{2^n}\), \(n\geq 1,\) we show that the solutions \(u_{{1/2^n}}\) correspond to the distribution of the \(n\)-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order \(\nu =\frac{2}{3^n}\), \(n\geq 1,\) is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that \(u_{\nu}\) coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions \(u_{\nu}\) and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.
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          Fractional processes: from Poisson to branching one

           ,  ,   (2010)
          Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z=N/ are found for both processes.
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            Author and article information

            Journal
            26 March 2013
            1303.6681
            10.1007/s11009-010-9207-6

            http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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            Methodology and Computing in Applied Probability, Vol. 14 (2), 383-403, 2012
            math.PR

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