Fractional immigration-death processes

Ascione, Giacomo; Leonenko, Nikolai; Pirozzi, Enrica

doi:10.1016/j.jmaa.2020.124768

Cited by 10 publications

(4 citation statements)

References 37 publications

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“…On the other hand, in Section 5, we study the properties of a time-changed stochastic SIR model. In the linear context, time-changed processes provide the direct non-local counterpart of their parent process, as one can see, for instance, in [22][23][24][25]63,64], and many others. In the non-linear context, as for the SIR model, things can go differently.…”

Section: Discussionmentioning

confidence: 99%

“…Concerning population models, one obtains analogous fractional-order stochastic models (or more general non-local models) by using time-changed continuous-time Markov chains, as one can see, for instance, in [22][23][24][25]. This fractionalization procedure leads to non-local stochastic models which are in some sense the counterpart of non-local deterministic ones.…”

Section: Introductionmentioning

confidence: 99%

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On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Ascione

2020

Mathematics

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Fractional-order epidemic models have become widely studied in the literature. Here, we consider the generalization of a simple SIR model in the context of generalized fractional calculus and we study the main features of such model. Moreover, we construct semi-Markov stochastic epidemic models by using time changed continuous time Markov chains, where the parent process is the stochastic analog of a simple SIR epidemic. In particular, we show that, differently from what happens in the classic case, the deterministic model does not coincide with the large population limit of the stochastic one. This loss of fluid limit is then stressed in terms of numerical examples.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Ascione

2020

Mathematics

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“…In both cases, the stochastic representation of such solutions is given by means of time-changed Pearson diffusions (with inverse stable subordinators). Similar strategies have been shown to work for lattice approximation of fractional Pearson diffusions: in [7] the spectral decomposition of a fractional immigration-death process (that is the lattice approximation of the Ornstein-Uhlenbeck process) is presented. On the other hand, subordinated Pearson diffusions (in particular the subordinated Jacobi process) have been shown to be useful tools to obtain large deviation principles (see [26]).…”

Section: Introductionmentioning

confidence: 99%

Time-Non-Local Pearson Diffusions

2021

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In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.

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“…This is done, for instance, by composing a parent Markov process with the inverse of a subordinator. In the specific case of the α-stable subordinator, one refers to such a new process as a fractional version of the parent one, due to the link between this time-change procedure and fractional calculus, as one can see from [11][12][13][14][15][16][17], to cite some of the several works on the topic. Together with a purely mathematical interest, let us stress that this procedure leads also to some interesting applications (see, for instance, [18] in economics, [19] in computing and [20] in computational neurosciences).…”

Section: Introductionmentioning

confidence: 99%

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

2022

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Several queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation.

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Fractional immigration-death processes

Abstract: Ascio n e, Gi a c o m o, Le o n e n k o, My kol a a n d Pi r ozzi, E n ric a 2 0 2 1. F r a c tio n al i m mi g r a tio n-d e a t h p r o c e s s e s . Jou r n al of M a t h e m a ti c al An alysis a n d Applic a tio n s 4 9 5 (2) ,

Cited by 10 publications

References 37 publications

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

On the Construction of Some Deterministic and Stochastic Non-Local SIR Models

Time-Non-Local Pearson Diffusions

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

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