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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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
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
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
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
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
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TL;DR: In this paper, a martingale characterization for the Fractional Poisson process on the plane is given, and the authors extend this result to the mixed-fractional poisson process and show that this process is the solution of a system of fractional differential-difference equations.
Abstract: We present new properties for the Fractional Poisson process and the Fractional
Poisson field on the plane. A martingale characterization for Fractional Poisson
processes is given. We extend this result to Fractional Poisson fields, obtaining some
other characterizations. The fractional differential equations are studied. We consider
a more general Mixed-Fractional Poisson process and show that this process is the
stochastic solution of a system of fractional differential-difference equations. Finally,
we give some simulations of the Fractional Poisson field on the plane.
30 citations
TL;DR: A correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes is focused on.
Abstract: Anomalous transport is usually described either by models of continuous time random walks (CTRW) or, otherwise by fractional Fokker-Planck equations (FFPE). The asymptotic relation between properly scaled CTRW and fractional diffusion process has been worked out via various approaches widely discussed in literature. Here, we focus on a correspondence between CTRWs and time and space fractional diffusion equation stemming from two different methods aimed to accurately approximate anomalous diffusion processes. One of them is the Monte Carlo simulation of uncoupled CTRW with a L\'evy $\alpha$-stable distribution of jumps in space and a one-parameter Mittag-Leffler distribution of waiting times. The other is based on a discretized form of a subordinated Langevin equation in which the physical time defined via the number of subsequent steps of motion is itself a random variable. Both approaches are tested for their numerical performance and verified with known analytical solutions for the Green function of a space-time fractional diffusion equation. The comparison demonstrates trade off between precision of constructed solutions and computational costs. The method based on the subordinated Langevin equation leads to a higher accuracy of results, while the CTRW framework with a Mittag-Leffler distribution of waiting times provides efficiently an approximate fundamental solution to the FFPE and converges to the probability density function of the subordinated process in a long-time limit.
30 citations
TL;DR: In this paper, the authors studied the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinators, which they call TCFPP-I and TC FPP-II, respectively.
Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.
28 citations
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TL;DR: In this paper, the authors investigated the main properties of high-frequency exchange rate data in the setting of stochastic subordination and stable modeling, focusing on heavy-tailedness and long memory, together with their dependence on the sampling period.
Abstract: We investigate the main properties of high-frequency exchange rate data in the setting of stochastic subordination and stable modeling, focusing on heavy-tailedness and long memory, together with their dependence on the sampling period. We show that the the instrinsic time process exhibits strong long-range dependence and has increments well described by a Weibull law, while the return series in intrinsic time has weak long memory and is well approximated by a stable Levy motion. We also show that the stable domain of attraction offers a good fit to the returns in physical time, which leads us to consider as a realistic model for exchange rate data a process subordinated to an alpha-stable Levy motion (possibly fractional stable) by a long-memory intrinsic time process with Weibull distributed increments.
27 citations
TL;DR: The first-passage-time processes of the anomalous diffusion on the self-similar curves in two dimensions are studied and natural parametrized subordinated Schramm-Loewner evolution (NS-SLE) is defined as a mathematical tool that can model diffusion on fractal curves.
Abstract: We study the first-passage-time processes of the anomalous diffusion on the self-similar curves in two dimensions. The scaling properties of the mean-square displacement and mean first passage time of the fractional Brownian motion and subordinated walk on the different fractal curves (loop-erased random walk, harmonic explorer, and percolation front) are derived. We also define natural parametrized subordinated Schramm-Loewner evolution (NS-SLE) as a mathematical tool that can model diffusion on fractal curves. The scaling properties of the mean-square displacement and mean first passage time for NS-SLE are obtained by numerical means.
27 citations