Counting processes with Bernštein intertimes and random jumps

Orsingher, Enzo; Toaldo, Bruno

doi:10.1017/s0021900200113063

Cited by 21 publications

(50 citation statements)

References 13 publications

(7 reference statements)

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“…In a recent paper (see [13]), Orsingher and Toaldo have considered more general counting processes with Bernstein intertimes and random jumps. In this section we also show that the space-fractional Poisson process is a special case of this wide family of counting processes, that admits an explicit and simple form for P {T k < ∞}.…”

Section: Counting Processes With Bernstein Intertimes: General Resultsmentioning

confidence: 99%

Some probabilistic properties of fractional point processes

Garra

Orsingher

Scavino

2017

Stochastic Analysis and Applications

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Abstract. This paper studies the first hitting times of generalized Poisson processes N f (t), related to Bernstein functions f . For the space-fractional Poisson processes, N α (t), t > 0 (corresponding to f = x α ), the hitting probabilities P {T α k < ∞} are explicitly obtained and analyzed. The processes N f (t) are time-changed Poisson processes N (H f (t)) with subordinators H f (t) and here we study N n j=1 H fj (t) and obtain probabilistic features of these extended counting processes. A section of the paper is devoted to processes of the form N (G H,ν (t)) where G H,ν (t) are generalized grey Brownian motions. This involves the theory of time-dependent fractional operators of the McBride form. While the time-fractional Poisson process is a renewal process, we prove that the space-time Poisson process is no longer a renewal process.

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Section: Counting Processes With Bernstein Intertimes: General Resultsmentioning

confidence: 99%

Some probabilistic properties of fractional point processes

Garra

Orsingher

Scavino

2017

Stochastic Analysis and Applications

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“…There is a recent wide literature about fractional generalizations of Poisson processes (see e.g. [4,5,15,21,[23][24][25][26][27]) and we should explain in which sense we can speak about fractionality in the last case. The relationship of this kind of distributions with fractional calculus, first considered by Beghin and Orsingher in [3], is given by the fact that the probability generating function…”

Section: Generalized Com-poisson Processes Involving the α-Mittag-lefmentioning

confidence: 99%

A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability

2018

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Abstract:In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.

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“…The processes N(H f (t)), t ≥ 0, have positive integervalued jumps whose distribution can be expressed in terms of the corresponding Bernštein function f . In the papers [17], [8] the distributional properties, hitting times and governing equations for such processes were derived and specified for several choices of H f (t) (some particular cases can be also found in [16], [7], [10], [12]).…”

Section: Introductionmentioning

confidence: 99%