Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

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...

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...

Abstract: The thesis is devoted to non-stationary point process models as generalizations of the standard homogeneous Poisson process. The work can be divided in two parts. In the first part, we introduce a fractional non-homogeneous Poisson process (FNPP) by applying a random time change to the standard Poisson process. We characterize the FNPP by deriving its non-local governing equation. We further compute moments and covariance of the process and discuss the distribution of the arrival times. Moreover, we give both finite-dimensional and functional limit theorems for the FNPP and the corresponding fractional non-homogeneous compound Poisson process. The limit theorems are derived by using martingale methods, regular variation properties and Anscombe's theorem. Eventually, some of the limit results are verified via a Monte-Carlo simulation. In the second part, we analyze statistical point process models for durations between trades recorded in financial high-frequency trading data. We consider parameter settings for models which are non-stationary or very close to non-stationarity which is quite typical for estimated parameter sets of models fitted to financial data. Simulation, parameter estimation and in particular model selection are discussed for the following three models: a non-homogeneous normal compound Poisson process, the exponential autoregressive conditional duration model (ACD) and a Hawkes process model. In a Monte-Carlo simulation, we test the performance of the following information criteria for model selection: Akaike's information criterion, the Bayesian information criterion and the Hannan-Quinn information criterion. We are particularly interested in the relation between the rate of correct model selection and the underlying sample size. Our numerical results show that the model selection for the compound Poisson type model works best for small parameter numbers. Moreover, the results for Hawkes processes confirm the theoretical asymptotic distributions of model selection whereas for the ACD model the model selection exhibits adverse behavior in certain cases.

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.

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...

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.

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

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."

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