Heavy-tailed fractional Pearson diffusions

Leonenko, Nikolai; Papić, Ivan; Sikorskii, Alla; Šuvak, Nenad

doi:10.1016/j.spa.2017.03.004

Cited by 13 publications

(14 citation statements)

References 52 publications

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“…Develop a more sophisticated model combining stochastic differential calculus [23], fractional diffusion [24] and elements of direct simulation (the models of fractional diffusion will be used to understand the effects of slowing down of the epidemic).…”

Section: Consequences For the Mortality And Impact On Nhsmentioning

confidence: 99%

Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19

Zhigljavsky

Whitaker

Fesenko

et al. 2020

Preprint

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Coronavirus COVID-19 spreads through the population mostly based on social contact. To gauge the potential for widespread contagion, to cope with associated uncertainty and to inform its mitigation, more accurate and robust modelling is centrally important for policy making.We provide a flexible modelling approach that increases the accuracy with which insights can be made. We use this to analyse different scenarios relevant to the COVID-19 situation in the UK. We present a stochastic model that captures the inherently probabilistic nature of contagion between population members. The computational nature of our model means that spatial constraints (e.g., communities and regions), the susceptibility of different age groups and other factors such as medical pre-histories can be incorporated with ease. We analyse different possible scenarios of the COVID-19 situation in the UK. Our model is robust to small changes in the parameters and is flexible in being able to deal with different scenarios.This approach goes beyond the convention of representing the spread of an epidemic through a fixed cycle of susceptibility, infection and recovery (SIR). It is important to emphasise that standard SIR-type models, unlike our model, are not flexible enough and are also not stochastic and hence should be used with extreme caution. Our model allows both heterogeneity and inherent uncertainty to be incorporated. Due to the scarcity of verified data, we draw insights by calibrating our model using parameters from other relevant sources, including agreement on average (mean field) with parameters in SIR-based models.We use the model to assess parameter sensitivity for a number of key variables that characterise the COVID-19 epidemic. We also test several control parameters with respect to their influence on the severity of the outbreak. Our analysis shows that due to inclusion of spatial heterogeneity in the population and the asynchronous timing of the epidemic across different areas, the severity of the epidemic might be lower than expected from other models.We find that one of the most crucial control parameters that may significantly reduce the severity of the epidemic is the degree of separation of vulnerable people and people aged 70 years and over, but note also that isolation of other groups has an effect on the severity of the epidemic. It is important to remember that models are there to advise and not to replace reality, and that any action should be coordinated and approved by public health experts with experience in dealing with epidemics.The computational approach makes it possible for further extensive scenario-based analysis to be undertaken. This and a comprehensive study of sensitivity of the model to different parameters defining COVID-19 and its development will be the subject of our forthcoming paper. In that paper, we shall also extend the model where we will consider different probabilistic scenarios for infected people with mild and severe cases.

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Section: Consequences For the Mortality And Impact On Nhsmentioning

confidence: 99%

Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19

Zhigljavsky

Whitaker

Fesenko

et al. 2020

Preprint

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“…These diffusions are very well studied and widely applied, e.g., in financial practice. The properties of these diffusions can be found in the classical book [14], while more recent developments relying on spectral representation of their transition densities are covered in [18], [17], [19], [20] and [21]. Heavy-tailed Pearson diffusions have not yet found their wide applications, in part due to complex properties of the spectrum of their infinitesimal generators (3.1).…”

Section: Pearson Diffusionsmentioning

confidence: 99%

“…In [18] the spectral representations for the transition densities of non-heavy-tailed fPDs (OU, CIR, Jacobi) were obtained. Namely, it has been shown that the series Spectral representations of transition densities for fractional reciprocal gamma and Fisher-Snedecor diffusions were obtained in [19] using the asymptotic properties of confluent and Gauss hypergeometric functions (see [7] and [5]) related to the continuous part of the spectrum of the infinitesimal generator of the corresponding non-fractional Pearson diffusion. Here we point out that the case of the spectral representation of Student diffusion, having absolutely continuous part of the spectrum of multilplicity two, is still not completely resolved.…”

Section: Fractional Pearson Diffusionsmentioning

confidence: 99%

“…Spectral representations of the transition densities of fPDs can be used to obtain the explicit strong solutions of the corresponding fractional Cauchy problems for both backward and forward equations. For relevant results we refer to [18] for non-heavy-tailed fPDs and to [19] for the reciprocal gamma and Fisher-Snedecor fractional diffusions.…”

Section: Fractional Pearson Diffusionsmentioning

confidence: 99%

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Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology

Leonenko

Papić

Sikorskii

et al. 2020

Journal of Mathematical Analysis and Applications

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“…In this context, the spectral decomposition of Pearson diffusions has been revealed to be a quite powerful tool to explicitly express strong solutions of fractional diffusion equations. This lead to the introduction of fractional Pearson diffusions: in [43] the first spectral category was covered, in [44] the authors focused on the second spectral category, while, up to our knowledge, there were no known methods to extend the spectral decomposition method to the fractional case in the third spectral category. In the second case, to express strong solutions of fractional Kolmogorov equations, the semigroup approach presented in [13] has been used.…”

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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Heavy-tailed fractional Pearson diffusions

Cited by 13 publications

References 52 publications

Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19

Generic probabilistic modelling and non-homogeneity issues for the UK epidemic of COVID-19

Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology

Time-Non-Local Pearson Diffusions

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