scholarly journals The Fractional Poisson Process and the Inverse Stable Subordinator

2011 ◽  
Vol 16 (0) ◽  
pp. 1600-1620 ◽  
Author(s):  
Mark Meerschaert ◽  
Erkan Nane ◽  
P. Vellaisamy
Keyword(s):  
Poisson Process ◽  
Stable Subordinator ◽  
Fractional Poisson Process ◽  
Inverse Stable Subordinator

scholarly journals On the Fractional Poisson Process and the Discretized Stable Subordinator

Axioms ◽  
2015 ◽  
Vol 4 (3) ◽  
pp. 321-344 ◽  
Author(s):  
Rudolf Gorenflo ◽  
Francesco Mainardi
Keyword(s):  
Poisson Process ◽  
Stable Subordinator ◽  
Fractional Poisson Process

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals Optimal Layer Reinsurance for Compound Fractional Poisson Model

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Jiesong Zhang
Keyword(s):  
Poisson Process ◽  
Closed Form ◽  
Compound Poisson Process ◽  
Poisson Model ◽  
Adjustment Coefficient ◽  
Compound Poisson ◽  
Numerical Examples ◽  
Fractional Poisson Process ◽  
Reinsurance Treaty

In this paper, we study the optimal retentions for an insurer with a compound fractional Poisson surplus and a layer reinsurance treaty. Under the criterion of maximizing the adjustment coefficient, the closed form expressions of the optimal results are obtained. It is demonstrated that the optimal retention vector and the maximal adjustment coefficient are not only closely related to the parameter of the fractional Poisson process, but also dependent on the time and the claim intensity, which is different from the case in the classical compound Poisson process. Numerical examples are presented to show the impacts of the three parameters on the optimal results.

scholarly journals Counting processes with Bernštein intertimes and random jumps

2015 ◽  
Vol 52 (04) ◽  
pp. 1028-1044 ◽  
Author(s):  
Enzo Orsingher ◽  
Bruno Toaldo
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Negative Binomial ◽  
Counting Processes ◽  
Lévy Measure ◽  
Fractional Poisson Process ◽  
Independent Increments ◽  
Random Jumps ◽  
The Times ◽  
Passage Times

In this paper we consider point processes Nf (t), t &gt; 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure v. We obtain the general expression of the probability generating functions Gf of Nf , the equations governing the state probabilities pk f of Nf , and their corresponding explicit forms. We also give the distribution of the first-passage times Tk f of Nf , and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times of jumps with height lj () under the condition N(t) = k for all these special processes is investigated in detail.

scholarly journals Fractional Poisson Process: Long-Range Dependence and Applications in Ruin Theory

2014 ◽  
Vol 51 (3) ◽  
pp. 727-740 ◽  
Author(s):  
Romain Biard ◽  
Bruno Saussereau
Keyword(s):  
Poisson Process ◽  
Long Range ◽  
Risk Model ◽  
Nonstationary Process ◽  
Long Range Dependence ◽  
Insurance Company ◽  
Surplus Process ◽  
Fractional Poisson Process ◽  
Renewal Risk ◽  
Dependence Property

We study a renewal risk model in which the surplus process of the insurance company is modelled by a compound fractional Poisson process. We establish the long-range dependence property of this nonstationary process. Some results for ruin probabilities are presented under various assumptions on the distribution of the claim sizes.

Fractional Poisson process (II)☆

2006 ◽  
Vol 28 (1) ◽  
pp. 143-147 ◽  
Author(s):  
Xiao-Tian Wang ◽  
Zhi-Xiong Wen ◽  
Shi-Ying Zhang
Keyword(s):  
Poisson Process ◽  
Fractional Poisson Process
Sign in / Sign up

Export Citation Format

Share Document