Author

# A. Maheshwari

Other affiliations: Indian Institute of Technology Bombay

Bio: A. Maheshwari is an academic researcher from Indian Institute of Management Indore. The author has contributed to research in topic(s): Fractional Poisson process & Subordinator. The author has an hindex of 6, co-authored 14 publication(s) receiving 113 citation(s). Previous affiliations of A. Maheshwari include Indian Institute of Technology Bombay.

##### Papers

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TL;DR: In this article, a fractional negative binomial process FNBP was defined by replacing the Poisson process by a FPP in the gamma subordinated form of the NBP.

Abstract: In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Polya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

10 citations

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TL;DR: The space-time fractional Poisson process (STFPP) as mentioned in this paper is a generalization of the TFPP and the space fractional poisson process, defined by Orsingher and Poilto (2012).

Abstract: The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson...

5 citations

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TL;DR: In this paper, the Linnik Levy process (LLP) is proposed to model leptokurtic data with heavy-tailed behavior, and the authors give a step-by-step procedure of the parameters estimation and calibrate the parameters of the LLP with the Arconic Inc equity data taken from Yahoo finance.

Abstract: In the literature, the Linnik, Mittag-Leffler, Laplace and asymmetric Laplace distributions are the most known examples of geometric stable distributions. The geometric stable distributions are especially useful in the modeling of leptokurtic data with heavy-tailed behavior. They have found many interesting applications in the modeling of several physical phenomena and financial time-series. In this paper, we define the Linnik Levy process (LLP) through the subordination of symmetric stable Levy motion with gamma process. We discuss main properties of LLP like probability density function, Levy measure and asymptotic forms of marginal densities. We also consider the governing fractional-type Fokker–Planck equation. To show practical applications, we simulate the sample paths of the introduced process. Moreover, we give a step-by-step procedure of the parameters estimation and calibrate the parameters of the LLP with the Arconic Inc equity data taken from Yahoo finance. Further, some extensions of the introduced process are also discussed.

4 citations

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TL;DR: In this paper, the authors studied some extensions of the Poisson process of order $i$ for different forms of weights and also with the time-changed versions, with Bern\v stein subordinator playing the role of time.

Abstract: The Poisson process of order $i$ is a weighted sum of independent Poisson processes and is used to model the flow of clients in different services. In the paper below we study some extensions of this process, for different forms of the weights and also with the time-changed versions, with Bern\v stein subordinator playing the role of time. We focus on the analysis of hitting times of these processes obtaining sometimes explicit distributions. Since all the processes examined display a similar structure with multiple upward jumps sometimes they can skip all states with positive probability even on infinitely long time span.

1 citations

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TL;DR: In this paper, the authors studied the non-homogeneous space-time fractional Poisson process (NSTFPP), a generalization of the time fractional poisson process.

Abstract: The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto in \cite{sfpp}, is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson process (SFPP). We study the fractional generalization of the non-homogeneous Poisson process and call it the non-homogeneous space-time fractional Poisson process (NSTFPP). We compute their {\it pmf} and generating function and investigate the associated differential equation. The limit theorems and the law of iterated logarithm for the NSTFPP process are studied. We study the distributional properties, the asymptotic expansion of the correlation function of the non-homogeneous time fractional Poisson process (NTFPP) and subsequently investigate the long-range dependence (LRD) property of a special NTFPP. We investigate the limit theorem and the LRD property for the fractional non-homogeneous Poisson process (FNPP), studied by Leonenko et. al. (2016). Finally, we present some simulated sample paths of the NSTFPP process.

1 citations

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28 Aug 2011

TL;DR: In this paper, it was shown that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional poisson process with Mittag-Leffler waiting times.

Abstract: The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve
a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a
fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional
diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered
fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

49 citations

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TL;DR: In this paper, the authors considered point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν.

Abstract: In this paper we consider point processes Nf(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernstein functions f with Levy measure ν. We obtain the general expression of the probability generating functions Gf of Nf, the equations governing the state probabilities pkf of Nf, and their corresponding explicit forms. We also give the distribution of the first-passage times Tkf of Nf, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τjlj of jumps with height lj (∑j=1rlj = k) under the condition N(t) = k for all these special processes is investigated in detail.

38 citations

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TL;DR: In this article, a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whose probabilistic ability to satisfy a suitablesystemoffractionaldifference-differential equations is considered.

Abstract: We consider a fractional counting process with jumps of amplitude 1,2,...,k, withk∈N, whoseprobabilitiessatisfy a suitablesystemoffractionaldifference-differential equations. We obtain the moment generating function and the probability law of the result- ing process in terms of generalized Mittag-Leffler functions. We also discuss two equiv- alent representations both in terms of a compound fractional Poisson process and of a subordinator governed by a suitable fractional Cauchy problem. The first occurrence time of a jump of fixed amplitude is proved to have the same distribution as the waiting time of the first event of a classical fractional Poisson process, this extending a well-known property of the Poisson process. When k = 2 we also express the distribution of the first passage time of the fractional counting process in an integral form. Finally, we show that the ratios given by the powers of the fractional Poisson process and of the countingprocess over their means tend to 1 in probability.

22 citations

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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.

18 citations