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Lévy processes and infinitely divisible distributions

01 Jan 2013-
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.
Citations
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Journal ArticleDOI
TL;DR: In this article, a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Levy process is proposed to calculate the conditional characteristic function of integrals driven by MG-FLPs, leading to important predictions concerning multivariate fractional Brownian motion, fractional subordinators and general fractional stochastic differential equations.
Abstract: Molchan-Golosov fractional Levy processes (MG-FLPs) are introduced by way of a multivariate componentwise Molchan-Golosov transformation based on an n-dimensional driving Levy process. Using results of fractional calculus and infinitely divisible distributions, we are able to calculate the conditional characteristic function of integrals driven by MG-FLPs. This leads to important predictions concerning multivariate fractional Brownian motion, fractional subordinators, and general fractional stochastic differential equations. Examples are the fractional Levy Ornstein-Uhlenbeck and Cox-Ingersoll-Ross models. As an application we present a fractional credit model with a long range dependent hazard rate and calculate bond prices.

14 citations

Journal ArticleDOI
TL;DR: In this paper, the authors considered the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain and two observables, τ (∙) and V(∙), corresponding to the renewal times and jump sizes.
Abstract: In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn, n ≥ 0} and two observables, τ(∙) and V(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn, n ≥ 0} is a sequence of independent and identically distributed random variables.

14 citations

Posted Content
TL;DR: In this paper, the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups have been derived in terms of a new family of power series which includes modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function.
Abstract: We provide the increasing eigenfunctions associated to spectrally negative self-similar Feller semigroups, which have been introduced by Lamperti. These eigenfunctions are expressed in terms of a new family of power series which includes, for instance, the modified Bessel functions of the first kind and some generalizations of the Mittag-Leffler function. Then, we show that some specific combinations of these functions are Laplace transforms of self-decomposable or infinitely divisible distributions concentrated on the positive line with respect to the main argument, and, more surprisingly, with respect to a parameter of the process. In particular, this generalizes a result of Hartman (1976) obtained for the increasing solution of the Bessel differential equation. Finally, we compute, for some cases, the associated decreasing eigenfunctions and derive the Laplace transform of the exponential functionals of some spectrally negative Levy processes with a negative first moment.

14 citations


Cites background or methods from "Lévy processes and infinitely divis..."

  • ...The last statement is a straightforward consequence of the property of the law of additive processes, see Sato [32], Chapter 3.16....

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  • ...The last statement is a straightforward consequence of the property of the law of additive processes, see Sato [32], Chapter 3....

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  • ...We refer to the monographs of Sato [32] and Steutel and van Harn [33] for an excellent account on these sets of probability measures....

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Journal ArticleDOI
TL;DR: In this paper, a continuous-time MA (moving average) process (Xt)t ≥ 0 sampled at an equally spaced time grid, where the grid distance Δ > 0 is fixed and n denotes the number of observations, was investigated in the frequency domain.
Abstract: In this paper we investigate a continuous-time MA (moving average) process (Xt)t≥0 sampled at an equally spaced time grid {Δ,2Δ, …, nΔ}, where the grid distance Δ > 0 is fixed and n denotes the number of observations, in the frequency domain. We derive for the process (XkΔ)k∈ℕ with finite second moments the asymptotic behavior of the periodogram and of the lag-window spectral density estimator. The periodogram is not a consistent estimator for the spectral density of (XkΔ)k∈ℕ. Different periodogram frequencies are asymptotically independent and exponentially distributed like for ARMA processes in discrete time. This result is basic for frequency bootstraps. In contrast, the lag-window spectral density estimator is a consistent estimator for the spectral density of (XkΔ)k∈ℕ and moreover, it is asymptotically normally distributed.

14 citations

Journal ArticleDOI
TL;DR: In this paper, the dependence between the jumps is modeled by a so-called Pareto Levy measure, which is a natural standardization in the context of regular variation, and the dependence structure is analyzed in terms of spectral density and the tail integral.
Abstract: We consider regular variation of a Levy process X := ( X t ) t ≥0 in with Levy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Levy measures to those of a standard 1-stable Levy process, we decouple the marginal Levy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Levy measure , which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Levy measures and the Pareto Levy measure. Moreover, we define upper and lower tail dependence coefficients for the Levy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Levy measures.

14 citations


Cites methods from "Lévy processes and infinitely divis..."

  • ...3 of [26] that the 1-homogeneous PLM as a 1-stable Lévy measure has, for all B ∈ B(R), the representation (B) = ∫ S ∫ ∞ 0 1B(rφ)r −2 drμ̃S(dφ), where S denotes the unit sphere in R ....

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  • ...For details and more background on Lévy processes, we refer the reader to [26]....

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References
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BookDOI
01 Jan 2014
TL;DR: In this article, Kloeden, P., Ombach, J., Cyganowski, S., Kostrikin, A. J., Reddy, J.A., Pokrovskii, A., Shafarevich, I.A.
Abstract: Algebra and Famous Inpossibilities Differential Systems Dumortier.: Qualitative Theory of Planar Jost, J.: Dynamical Systems. Examples of Complex Behaviour Jost, J.: Postmodern Analysis Jost, J.: Riemannian Geometry and Geometric Analysis Kac, V.; Cheung, P.: Quantum Calculus Kannan, R.; Krueger, C.K.: Advanced Analysis on the Real Line Kelly, P.; Matthews, G.: The NonEuclidean Hyperbolic Plane Kempf, G.: Complex Abelian Varieties and Theta Functions Kitchens, B. P.: Symbolic Dynamics Kloeden, P.; Ombach, J.; Cyganowski, S.: From Elementary Probability to Stochastic Differential Equations with MAPLE Kloeden, P. E.; Platen; E.; Schurz, H.: Numerical Solution of SDE Through Computer Experiments Kostrikin, A. I.: Introduction to Algebra Krasnoselskii, M.A.; Pokrovskii, A.V.: Systems with Hysteresis Kurzweil, H.; Stellmacher, B.: The Theory of Finite Groups. An Introduction Lang, S.: Introduction to Differentiable Manifolds Luecking, D.H., Rubel, L.A.: Complex Analysis. A Functional Analysis Approach Ma, Zhi-Ming; Roeckner, M.: Introduction to the Theory of (non-symmetric) Dirichlet Forms Mac Lane, S.; Moerdijk, I.: Sheaves in Geometry and Logic Marcus, D.A.: Number Fields Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis Matoušek, J.: Using the Borsuk-Ulam Theorem Matsuki, K.: Introduction to the Mori Program Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 1 Mazzola, G.; Milmeister G.; Weissman J.: Comprehensive Mathematics for Computer Scientists 2 Mc Carthy, P. J.: Introduction to Arithmetical Functions McCrimmon, K.: A Taste of Jordan Algebras Meyer, R.M.: Essential Mathematics for Applied Field Meyer-Nieberg, P.: Banach Lattices Mikosch, T.: Non-Life Insurance Mathematics Mines, R.; Richman, F.; Ruitenburg, W.: A Course in Constructive Algebra Moise, E. E.: Introductory Problem Courses in Analysis and Topology Montesinos-Amilibia, J.M.: Classical Tessellations and Three Manifolds Morris, P.: Introduction to Game Theory Nikulin, V.V.; Shafarevich, I. R.: Geometries and Groups Oden, J. J.; Reddy, J. N.: Variational Methods in Theoretical Mechanics Øksendal, B.: Stochastic Differential Equations Øksendal, B.; Sulem, A.: Applied Stochastic Control of Jump Diffusions Poizat, B.: A Course in Model Theory Polster, B.: A Geometrical Picture Book Porter, J. R.; Woods, R.G.: Extensions and Absolutes of Hausdorff Spaces Radjavi, H.; Rosenthal, P.: Simultaneous Triangularization Ramsay, A.; Richtmeyer, R.D.: Introduction to Hyperbolic Geometry Rees, E.G.: Notes on Geometry Reisel, R. B.: Elementary Theory of Metric Spaces Rey, W. J. J.: Introduction to Robust and Quasi-Robust Statistical Methods Ribenboim, P.: Classical Theory of Algebraic Numbers Rickart, C. E.: Natural Function Algebras Roger G.: Analysis II Rotman, J. J.: Galois Theory Jost, J.: Compact Riemann Surfaces Applications ́ Introductory Lectures on Fluctuations of Levy Processes with Kyprianou, A. : Rautenberg, W.; A Concise Introduction to Mathematical Logic Samelson, H.: Notes on Lie Algebras Schiff, J. L.: Normal Families Sengupta, J.K.: Optimal Decisions under Uncertainty Séroul, R.: Programming for Mathematicians Seydel, R.: Tools for Computational Finance Shafarevich, I. R.: Discourses on Algebra Shapiro, J. H.: Composition Operators and Classical Function Theory Simonnet, M.: Measures and Probabilities Smith, K. E.; Kahanpää, L.; Kekäläinen, P.; Traves, W.: An Invitation to Algebraic Geometry Smith, K.T.: Power Series from a Computational Point of View Smoryński, C.: Logical Number Theory I. An Introduction Stichtenoth, H.: Algebraic Function Fields and Codes Stillwell, J.: Geometry of Surfaces Stroock, D.W.: An Introduction to the Theory of Large Deviations Sunder, V. S.: An Invitation to von Neumann Algebras Tamme, G.: Introduction to Étale Cohomology Tondeur, P.: Foliations on Riemannian Manifolds Toth, G.: Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems Wong, M.W.: Weyl Transforms Xambó-Descamps, S.: Block Error-Correcting Codes Zaanen, A.C.: Continuity, Integration and Fourier Theory Zhang, F.: Matrix Theory Zong, C.: Sphere Packings Zong, C.: Strange Phenomena in Convex and Discrete Geometry Zorich, V.A.: Mathematical Analysis I Zorich, V.A.: Mathematical Analysis II Rybakowski, K. P.: The Homotopy Index and Partial Differential Equations Sagan, H.: Space-Filling Curves Ruiz-Tolosa, J. R.; Castillo E.: From Vectors to Tensors Runde, V.: A Taste of Topology Rubel, L.A.: Entire and Meromorphic Functions Weintraub, S.H.: Galois Theory

401 citations

Journal ArticleDOI
TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.

372 citations

Book ChapterDOI
TL;DR: In this article, the authors give an up-to-date account of the theory and applications of scale functions for spectrally negative Levy processes, including the first extensive overview of how to work numerically with scale functions.
Abstract: The purpose of this review article is to give an up to date account of the theory and applications of scale functions for spectrally negative Levy processes. Our review also includes the first extensive overview of how to work numerically with scale functions. Aside from being well acquainted with the general theory of probability, the reader is assumed to have some elementary knowledge of Levy processes, in particular a reasonable understanding of the Levy–Khintchine formula and its relationship to the Levy–Ito decomposition. We shall also touch on more general topics such as excursion theory and semi-martingale calculus. However, wherever possible, we shall try to focus on key ideas taking a selective stance on the technical details. For the reader who is less familiar with some of the mathematical theories and techniques which are used at various points in this review, we note that all the necessary technical background can be found in the following texts on Levy processes; (Bertoin, Levy Processes (1996); Sato, Levy Processes and Infinitely Divisible Distributions (1999); Kyprianou, Introductory Lectures on Fluctuations of Levy Processes and Their Applications (2006); Doney, Fluctuation Theory for Levy Processes (2007)), Applebaum Levy Processes and Stochastic Calculus (2009).

288 citations

Journal ArticleDOI
TL;DR: A closed formula for prices of perpetual American call options in terms of the overall supremum of the Lévy process, and a corresponding closed formulas for perpetual American put options involving the infimum of the after-mentioned process are obtained.
Abstract: Consider a model of a financial market with a stock driven by a Levy process and constant interest rate. A closed formula for prices of perpetual American call options in terms of the overall supremum of the Levy process, and a corresponding closed formula for perpetual American put options involving the infimum of the after-mentioned process are obtained. As a direct application of the previous results, a Black-Scholes type formula is given. Also as a consequence, simple explicit formulas for prices of call options are obtained for a Levy process with positive mixed-exponential and arbitrary negative jumps. In the case of put options, similar simple formulas are obtained under the condition of negative mixed-exponential and arbitrary positive jumps. Risk-neutral valuation is discussed and a simple jump-diffusion model is chosen to illustrate the results.

269 citations

01 May 2013
TL;DR: In this paper, the authors review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics.
Abstract: We review work on extreme events, their causes and consequences, by a group of European and American researchers involved in a three-year project on these topics. The review covers theoretical aspects of time series analysis and of extreme value theory, as well as of the deterministic modeling of extreme events, via continuous and discrete dynamic models. The applications include climatic, seismic and socio-economic events, along with their prediction. Two important results refer to (i) the complementarity of spectral analysis of a time series in terms of the continuous and the discrete part of its power spectrum; and (ii) the need for coupled modeling of natural and socio-economic systems. Both these results have implications for the study and prediction of natural hazards and their human impacts.

166 citations