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 paper , the authors consider convolution operators with nonsingular kernels and prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators.
Abstract: We consider here convolution operators, in the Caputo sense, with nonsingular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of Lévy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel’s parameters and, consequently, of the jumps’ density function.
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TL;DR: In this article, a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator is introduced and an integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained.
Abstract: We introduce and study a fractional version of the Skellam process of order $k$ by time-changing it with an independent inverse stable subordinator. We call it the fractional Skellam process of order $k$ (FSPoK). An integral representation for its one-dimensional distributions and their governing system of fractional differential equations are obtained. We derive the probability generating function, mean, variance and covariance of the FSPoK which are utilized to establish its long-range dependence property. Later, we considered two time-changed versions of the FSPoK. These are obtained by time-changing the FSPoK by an independent L\'evy subordinator and its inverse. Some distributional properties and particular cases are discussed for these time-changed processes.
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TL;DR: In this article , the authors defined a convoluted fractional Poisson process of order k (CFPPoK), which is governed by the discrete convolution operator in the system of fractional differential equations, and obtained its one-dimensional distribution by using the Laplace transform of its state probabilities.
Abstract: In this article, we define a convoluted fractional Poisson process of order k (CFPPoK), which is governed by the discrete convolution operator in the system of fractional differential equations. Next, we obtain its one-dimensional distribution by using the Laplace transform of its state probabilities. Various distributional properties, such as probability generating function, moment generating function and moments, are derived. A special case of CFPPoK, (say) convoluted Poisson process of order k (CPPoK) is studied and also established Martingale characterization for CPPoK. We further derive the covariance structure of CFPPoK and investigate the long-range dependence property.
TL;DR: The marginal probabilities, Lévy measures, governing difference-differential equations of the introduced processes, and the results generalize the Skellam process and running average of Poisson process in several directions.
Abstract: In this article, we introduce Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular we discuss space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Poisson 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 Skellam process and running average of Poisson process in several directions.
TL;DR: In this article , the generalized fractional Skellam process (GFSP) is considered by time-changing it with an independent inverse stable subordinator, and it is observed that the GFSP is a Skellham type version of the generalized fractional counting process (GFCP), which is a fractional variant of the GCP.
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 Lévy subordinator and its inverse. Some particular cases of these time-changed processes are discussed by considering specific Lévy subordinators.
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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
TL;DR: In this article, a general class of finite-variance distributions for price changes is described, and a member of this class, the lognormal-normal, is tested against previously proposed distributions for speculative price differences.
Abstract: S. Bochner's concept of a subordinate stochastic process is proposed as a model for speculative price series. A general class of finite-variance distributions for price changes is described, and a member of this class, the lognormal-normal, is tested against previously proposed distributions for speculative price differences. It is shown with both discrete Bayes' tests and Kolmogorov-Smirnov tests that finite-variance distributions subordinate to the normal fit cotton futures price data better than members of the stable family.
2,941 citations
TL;DR: A number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed, including Ornstein-Uhlenbeck type processes, superpositions of such processes and Stochastic volatility models in one and more dimensions.
Abstract: With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.
1,323 citations
TL;DR: In this article, the potential of the normal inverse Gaussian distribution and the Levy process for modeling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance, is discussed.
Abstract: The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mixing distribution. The distribution determines an homogeneous Levy process, and this process is representable through subordination of Brownian motion by the inverse Gaussian process. The canonical, Levy type, decomposition of the process is determined. As a preparation for developments in the latter part of the paper the connection of the normal inverse Gaussian distribution to the classes of generalized hyperbolic and inverse Gaussian distributions is briefly reviewed. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Levy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance. These areas of application imply a need for extending the inverse Gaussian Levy process so as to accommodate certain, frequently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-driven approach to state space modelling. At the end of the paper generalizations to multivariate settings are indicated.
998 citations