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Journal ArticleDOI

Time-changed Poisson processes of order k

TL;DR: In this article, the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse was studied, which they called TCPPoK-I and TCPPoK-II.
Abstract: In this article, we study the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse, which we call, respectively, as TCPPoK-I and TCPPoK-II, t...
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Journal ArticleDOI
22 Oct 2020-Entropy
TL;DR: In this article, the Skellam process of order k and its running average was introduced and the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes were derived.
Abstract: In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions.

12 citations

Journal ArticleDOI
TL;DR: In this article, the hitting probabilities of weighted Poisson processes and their subordinated versions with different intensities were studied. And the authors analyzed the hitting probability in different weights and gave an example in the case of subordination.

4 citations

Journal ArticleDOI
TL;DR: The generalized fractional counting process (GFCP) was introduced and studied by Di Crescenzo et al. as discussed by the authors , and its covariance structure is studied, using which its long-range dependence property is established.
Abstract: In this paper, we obtain additional results for a fractional counting process introduced and studied by Di Crescenzo et al. [8]. For convenience, we call it the generalized fractional counting process (GFCP). It is shown that the one-dimensional distributions of the GFCP are not infinitely divisible. Its covariance structure is studied, using which its long-range dependence property is established. It is shown that the increments of GFCP exhibit the short-range dependence property. Also, we prove that the GFCP is a scaling limit of some continuous time random walk. A particular case of the GFCP, namely, the generalized counting process (GCP), is discussed for which we obtain a limiting result and a martingale result and establish a recurrence relation for its probability mass function. We have shown that many known counting processes such as the Poisson process of order k, the Pólya-Aeppli process of order k, the negative binomial process and their fractional versions etc., are other special cases of the GFCP. An application of the GCP to risk theory is discussed.

4 citations

Journal ArticleDOI
01 May 2020
TL;DR: In this article, the compound Poisson processes of order $k$ (CPPoK) were introduced and its properties were discussed, using mixture of tempered stable subordinator and its right continuous inverse, the two subordinated CPPoK with various distributional properties were studied.
Abstract: In this article, the compound Poisson processes of order $k$ (CPPoK) is introduced and its properties are discussed. Further, using mixture of tempered stable subordinator (MTSS) and its right continuous inverse, the two subordinated CPPoK with various distributional properties are studied. It is also shown that space and tempered space fractional versions of CPPoK and PPoK can be obtained, which generalize the results in the literature.

3 citations

Journal ArticleDOI
TL;DR: In this paper, a fractional non-homogeneous Poisson Poisson process of order k and polya-aeppli Poisson Process of order K were characterized by deriving their non-local governing equations.
Abstract: We introduce two classes of point processes: a fractional non-homogeneous Poisson process of order k and a fractional non-homogeneous Polya-Aeppli process of order k: We characterize these processes by deriving their non-local governing equations. We further study the covariance structure of the processes and investigate the long-range dependence property.

2 citations

References
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MonographDOI
01 Jan 2009

902 citations

Journal ArticleDOI
TL;DR: This work tracks the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells and finds that the motion is subdiffusive, with an exponent that is robust to physiological changes, including the disruption of cytoskeletal elements.
Abstract: We track the motion of individual fluorescently labeled mRNA molecules inside live E. coli cells. We find that the motion is subdiffusive, with an exponent that is robust to physiological changes, including the disruption of cytoskeletal elements. By modifying the parameters of the RNA molecule and the bacterial cell, we are able to examine the possible mechanisms that can lead to this unique type of motion, especially the effect of macromolecular crowding. We also examine the implications of anomalous diffusion on the kinetics of bacterial gene regulation, in particular, how transcription factors find their DNA targets.

764 citations

Journal ArticleDOI
TL;DR: In this paper, the daily and weekly seasonality of foreign exchange volatility is modelled by introducing an activity variable, which is explained by a simple model of the changing and sometimes overlapping market presence of geographical components (East Asia, Europe, and America).

620 citations

Posted Content
TL;DR: The multifractal model of asset returns (MMARCH) as mentioned in this paper is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years.
Abstract: This paper presents the multifractal model of asset returns ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well-known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second. the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multi-scaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time finance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper.

479 citations

Posted Content
TL;DR: The multifractal model of asset returns (MMARCH) as discussed by the authors is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years.
Abstract: This paper presents the multifractal model of asset returns ("MMAR"), based upon the pioneering research into multifractal measures by Mandelbrot (1972, 1974). The multifractal model incorporates two elements of Mandelbrot's past research that are now well-known in finance. First, the MMAR contains long-tails, as in Mandelbrot (1963), which focused on Levy-stable distributions. In contrast to Mandelbrot (1963), this model does not necessarily imply infinite variance. Second. the model contains long-dependence, the characteristic feature of fractional Brownian Motion (FBM), introduced by Mandelbrot and van Ness (1968). In contrast to FBM, the multifractal model displays long dependence in the absolute value of price increments, while price increments themselves can be uncorrelated. As such, the MMAR is an alternative to ARCH-type representations that have been the focus of empirical research on the distribution of prices for the past fifteen years. The distinguishing feature of the multifractal model is multi-scaling of the return distribution's moments under time-rescalings. We define multiscaling, show how to generate processes with this property, and discuss how these processes differ from the standard processes of continuous-time finance. The multifractal model implies certain empirical regularities, which are investigated in a companion paper.

332 citations