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Generalized Fractional Counting Process
TL;DR: The generalized fractional counting process (GFCP) was introduced and studied by Di Crescenzo et al. as mentioned in this paper, 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. (2016). 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 exhibits 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, 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 Polya-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.
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TL;DR: In this paper, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator, and its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained.
Abstract: In this paper, we study a Skellam type variant of the generalized counting process (GCP), namely, the generalized Skellam process. Some of its distributional properties such as the probability mass function, probability generating function, mean, variance and covariance are obtained. Its fractional version, namely, the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator. It is observed that the GFSP is a Skellam type version of the generalized fractional counting process (GFCP) which is a fractional variant of the GCP. It is shown that the one-dimensional distributions of the GFSP are not infinitely divisible. An integral representation for its state probabilities is obtained. We establish its long-range dependence property by using its variance and covariance structure. Also, we consider two time-changed versions of the GFCP. These are obtained by time-changing the GFCP by an independent Levy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Levy subordinators.
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TL;DR: In this paper, the generalized fractional birth process (GFBP) was introduced and a non-exploding condition for it was derived, and the system of differential equations that governs its state probabilities was obtained.
Abstract: In this paper, we introduce a generalized birth process (GBP) which performs jumps of size $1,2,\dots,k$ whose rates depend on the state of the process at time $t\geq0$. We derive a non-exploding condition for it. The system of differential equations that governs its state probabilities is obtained. In this governing system of differential equations, we replace the first order derivative with Caputo fractional derivative to obtain a fractional variant of the GBP, namely, the generalized fractional birth process (GFBP). The Laplace transform of the state probabilities of this fractional variant is obtained whose inversion yields its one-dimensional distribution. It is shown that the GFBP is equal in distribution to a time-changed version of the GBP, and this result is used to obtain a non-exploding condition for it. A limiting case of the GFBP is considered in which jump of any size $j\ge1$ is possible. Also, we discuss a state dependent version of it.
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Book•
02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
TL;DR: In this paper, a Monte-Carlo analysis of stock market returns was conducted and it was found that not only there is substantially more correlation between absolute returns than returns themselves, but the power transformation of the absolute return also has quite high autocorrelation for long lags.
3,462 citations
TL;DR: A restatement in terms of set partitions can be proved easily in a few lines, as the authors shall see in Section 2, though it still requires a bit of work to pass from that form to the form in (1.1).
Abstract: (2002). The Curious History of Faa di Bruno's Formula. The American Mathematical Monthly: Vol. 109, No. 3, pp. 217-234.
574 citations
Book•
03 Oct 2003
TL;DR: In this paper, the authors present a list of well-known distributions notations and conventions, as well as prerequisites from probability and analysis list of distributions notation and conventions for stochastic processes.
Abstract: infinitely divisible distributions on the nonnegative integers infinitely divisible distributions on the nonnegative reals infinitely divisible distributions on the real line self-decomposability and stability infinite divisibility and mixtures infinite divisibility in stochastic processes. Appendices: prerequisites from probability and analysis list of well-known distributions notations and conventions.
488 citations