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Random-time processes governed by differential equations of fractional distributed order

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  • Beghin, L.

Abstract

We analyze here different types of fractional differential equations, under the assumption that their fractional order ν∈(0,1] is random with probability density n(ν). We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process N(t), t>0. We prove that, for a particular (discrete) choice of n(ν), it leads to a process with random time, defined as N(T∼ν1,ν2(t)),t>0. The distribution of the random time argument T∼ν1,ν2(t) can be expressed, for any fixed t, in terms of convolutions of stable-laws. The new process N(T∼ν1,ν2) is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of N(T∼ν1,ν2), as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see [16]).

Suggested Citation

  • Beghin, L., 2012. "Random-time processes governed by differential equations of fractional distributed order," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1314-1327.
  • Handle: RePEc:eee:chsofr:v:45:y:2012:i:11:p:1314-1327
    DOI: 10.1016/j.chaos.2012.07.001
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    1. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    2. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
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    Cited by:

    1. K. K. Kataria & M. Khandakar, 2021. "On the Long-Range Dependence of Mixed Fractional Poisson Process," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1607-1622, September.
    2. K. K. Kataria & M. Khandakar, 2022. "Generalized Fractional Counting Process," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2784-2805, December.
    3. Zhang, Meihui & Jia, Jinhong & Zheng, Xiangcheng, 2023. "Numerical approximation and fast implementation to a generalized distributed-order time-fractional option pricing model," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    4. Gajda, Janusz & Beghin, Luisa, 2021. "Prabhakar Lévy processes," Statistics & Probability Letters, Elsevier, vol. 178(C).
    5. L. Beghin & P. Vellaisamy, 2018. "Space-Fractional Versions of the Negative Binomial and Polya-Type Processes," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 463-485, June.
    6. Lee Jeonghwa, 2021. "Generalized Bernoulli process: simulation, estimation, and application," Dependence Modeling, De Gruyter, vol. 9(1), pages 141-155, January.
    7. Lee Jeonghwa, 2021. "Generalized Bernoulli process with long-range dependence and fractional binomial distribution," Dependence Modeling, De Gruyter, vol. 9(1), pages 1-12, January.

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