Asymptotic degeneracy and subdiffusivity

Yuan, Sida; Kawai, Reiichiro

doi:10.1088/1751-8121/ab69a5

“…When sample path generation via increments is possible, one important distinction between such a scheme and that of a Poisson truncation method is that in the latter, a terminal time T is fixed beforehand and cannot be extended during the simulation, whereas in the former, one can keep piling on as many increments as desired, possibly until some condition is fulfilled. An example of when this distinction is important is through a comparison between the settings of [105] and [18]. In the former, Poisson truncation of shot noise representation is used to generate sample paths of an inverse subordinator.…”

Section: Discussionmentioning

confidence: 99%

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

Yuan,

Kawai

2021

Preprint

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The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, Lévy processes, infinitely divisible processes and fields and Lévy-driven stochastic differential equations. Essential notions and results towards practical implementation are outlined, and summaries of simulation recipes are provided throughout along with numerical illustrations. Some future research directions are highlighted.

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“…When sample path generation via increments is possible, one important distinction between such a scheme and that of a Poisson truncation method is that in the latter, a terminal time T is fixed beforehand and cannot be extended during the simulation, whereas in the former, one can keep piling on as many increments as desired, possibly until some condition is fulfilled. An example of when this distinction is important is through a comparison between the settings of [106] and [18]. In the former, Poisson truncation of shot noise representation is used to generate sample paths of an inverse subordinator.…”

Section: Discussionmentioning

confidence: 99%

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

Yuan

¹

,

Kawai

²

2021

Self Cite

View full text Add to dashboard Cite

The ever-growing appearance of infinitely divisible laws and related processes in various areas, such as physics, mathematical biology, finance and economics, has fuelled an increasing demand for numerical methods of sampling and sample path generation. In this survey, we review shot noise representation with a view towards sampling infinitely divisible laws and generating sample paths of related processes. In contrast to many conventional methods, the shot noise approach remains practical even in the multidimensional setting. We provide a brief introduction to shot noise representations of infinitely divisible laws and related processes, and discuss the truncation of such series representations towards the simulation of infinitely divisible random vectors, Lévy processes, infinitely divisible processes and fields and Lévy-driven stochastic differential equations. Essential notions and results towards practical implementation are outlined, and summaries of simulation recipes are provided throughout along with numerical illustrations. Some future research directions are highlighted.

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“…. We refer the reader to [53] for more detail about the asymptotic degeneracy of inverse subordinators.…”

Section: Theorem 33 (Regularity)mentioning

confidence: 99%

“…In the literature, the inverse stable subordinator has been investigated from a variety of perspectives for which we refer the reader to [61,62,44,63,53], to name just a few.…”

Section: A1 Inverse Stable Subordinatormentioning

confidence: 99%

Dimension dependent properties of subdiffusions in damping force fields from an inference perspective

He¹,

Kawai²

2022

Phys. Scr.

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0

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We investigate the fractional Fokker-Planck equation subject to a damping force with an emphasis on its dimension dependent properties. We reveal a variety of surprising properties of its solution through the lens of the probability density function of the corresponding stochastic process with nonlinear mean square displacements, such as existence, singularity, regularity, modality, stationarity and second-order structure, which are largely dependent on the dimension and the random clock. Taking into account that the trajectory information is most often collected from multidimensional systems, the discovered facts have the potential to play important roles as key foundations and alerts for inference, model identification and prediction, when departing from the well-understood univariate framework.

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Asymptotic degeneracy and subdiffusivity

Cited by 5 publications

References 23 publications

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

Numerical aspects of shot noise representation of infinitely divisible laws and related processes

Dimension dependent properties of subdiffusions in damping force fields from an inference perspective

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