Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse
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.
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TL;DR: In this paper, the authors define the delayed Levy-driven continuous-time autoregressive process via the inverse of the stable subordinator and derive correlation structure for the observed non-stationary delayed Levy drive.
Abstract: We define the delayed Levy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Levy-dri...
3 citations
Cites methods from "Fractional Poisson Process Time-Cha..."
...Long-range dependence property in the form of Definition 5.1 was first used in Maheshwari & Vellaisamy (2016) and Maheshwari & Vellaisamy (2017)....
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TL;DR: In this paper, the authors studied the Poisson process of order k (PPoK) time-changed with an independent Levy subordinator and its inverse, which they call respectively, as TCPPoK-I and TCPPOK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCP-I.
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, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.
2 citations
Cites methods from "Fractional Poisson Process Time-Cha..."
...We refer to Algorithm 2–5 from [17] to generate sample paths of the gamma and the inverse Gaussian subordinator and their right-continuous inverses....
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...(c) Simulate the values WðtiÞ, 1 i n, of the subordinator (inverse subordinator) at t1, :::tn, using the Algorithm 2–5 of [17] for respective subordinator (inverse subordinator)....
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...Therefore, the index of dispersion IðtÞ :1⁄4 Var1⁄2Qð1Þ f ðtÞ =E1⁄2Qð1Þ f ðtÞ (see [17] for more details) is greater than 1....
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...It is defined as EGðtÞ 1⁄4 inffr 0 : GðrÞ> tg, t 0: The mean of EGðtÞ is given by (see [17, 45]) MðtÞ 1⁄4 E EGðtÞ 1⁄2 1⁄4 Cð2Þ sðd ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2sþ c2 p c Þ :...
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...The Laplace Transform (LT) of pth moment of Ef ðtÞ is given by (see [17])...
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TL;DR: In this paper, the authors studied the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution and found that the FP sum contains the following terms:
Abstract: In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains th...
2 citations
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TL;DR: In this article, the authors introduced fractional Poisson felds of order k in n-dimensional Euclidean space and derived the marginal probabilities, governing difference-differential equations of the introduced processes.
Abstract: In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poisson process of order k, allows on average infinite number of arrivals in any interval. We derive the marginal probabilities, governing difference-differential equations of the introduced processes. We also provide Watanabe martingale characterization for some time-changed Poisson processes.
2 citations
Additional excerpts
...Maheshwari and Vellaisamy [22, 23])....
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TL;DR: In this paper, a time-changed version of the space-time fractional Poisson process (STFPP) by time changing it by an independent Levy subordinator with finite moments of any
Abstract: In this paper, we introduce and study a time-changed version of the space-time fractional Poisson process (STFPP) by time changing it by an independent Levy subordinator with finite moments of any
2 citations
Cites background or result from "Fractional Poisson Process Time-Cha..."
...For a1⁄4 1, the TCFPP reduces to the TCFPP-I, fN 1,b f ðtÞgt 0, 0 < b 1, (see [9])....
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...Recently, Maheshwari and Vellaisamy [9] introduced and studied two processes by time-changing the TFPP with an independent L evy subordinator and its inverse....
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...Cðkaþ 1 nÞ , which coincides with the pmf of the space fractional negative binomial process (SFNBP) introduced and studied by Beghin and Vellaisamy [28]....
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...The results obtained in this paper generalize and complement the results of [9, 11] and [8]....
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...For a1⁄4 1, the process defined in (8) reduces to TCFPP-I (see [9])....
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References
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30 Dec 2003
TL;DR: In this article, the authors provide a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists.
Abstract: WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Levy processes are beyond their reach.Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by their use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations.Topics covered in this book include: jump-diffusion models, Levy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms.This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable new set of tools for modelling market fluctuations.
3,210 citations
Book•
01 Jan 2004TL;DR: In this paper, the authors present a general theory of Levy processes and a stochastic calculus for Levy processes in a direct and accessible way, including necessary and sufficient conditions for Levy process to have finite moments.
Abstract: Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs
2,908 citations
Book•
01 Jan 1966
TL;DR: This monograph is intended as a survey of some of the problems in theoretical statistics that stem from this sort of data, and has tried to give a simple description, with numerical examples, of the main methods that have been proposed.
Abstract: Observations in the form of point events occurring in a continuum, space or time, arise in many fields of study. In writing this monograph on statistical techniques for dealing with such data, we have three objectives. First, we have tried to give a simple description, with numerical examples, of the main methods that have been proposed. We hope that by concentrating on the examples the applied statistician with a limited inclination for theory will find something of practical value in the monograph. Second, the monograph is intended as a survey, necessarily incomplete, of some of the problems in theoretical statistics that stem from this sort of data. A number of specialized subjects have, however, been dealt with only briefly, the main emphasis being placed on the problem of examining the structure of a series of events. Finally, we hope that the monograph will be of use to teachers and students of statistics, as illustrating applications of a range of tech niques in theoretical statistics. We are extremely grateful to the International Business Machines Corporation for providing programming assistance and a large amount of computer time. We wish to thank particularly Mr A."
1,993 citations
01 Jan 1971
TL;DR: In this article, a linear operator of order functions of order (1.2) is defined and an operator of fractional integration is employed to prove results on the solutions of the integral equation.
Abstract: is an entire function of order $({\rm Re}\alpha)^{-1}$ and contains several well-known special functions as particular cases. We define a linear operator $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ on a space $L$ of functions by the integral in (1.2) and employ an operator of fractional integration $I^{\mu}$ : $L\rightarrow L$ to prove results on $\mathfrak{C}(\alpha, \beta;\rho;\lambda)$ ; these results are subsequently used to discuss theorems on the solutions of (1.2). The technique used can be apPlied to obtain analogous results on the integral equation
822 citations