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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.
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TL;DR: In this article, the authors present a dialogue that appeals as much as possible to path decompositions for stable Lévy processes in one and higher dimensions, with a focus on the Riesz-Bogdan-Żak transformation.
Abstract: Around the 1960s a celebrated collection of papers emerged offering a number of explicit identities for the class of isotropic Lévy stable processes in one and higher dimensions; these include, for example, the lauded works of Ray (1958); Widom (1961); Rogozin (1972) (in one dimension) and Blumenthal et al. (1961); Getoor (1966); Port (1969) (in higher dimension), see also more recently Byczkowski et al. (2009); Luks (2013). Amongst other things, these results nicely exemplify the use of standard Riesz potential theory on the unit open ball Bd := {x ∈ R : |x| < 1}, R\\Bd and the sphere Sd−1 := {x ∈ R : |x| = 1} with the, then, modern theory of potential analysis for Markov processes. Following initial observations of Lamperti (1972), with the occasional sporadic contributions such as Kiu (1980); Vuolle-Apiala and Graversen (1986); Graversen and Vuolle-Apiala (1986), an alternative understanding of Lévy stable processes through the theory of self-similar Markov processes has prevailed in the last decade or more. This point of view offers deeper probabilistic insights into some of the aforementioned potential analytical relations; see for example Bertoin and Yor (2002); Bertoin and Caballero (2002); Caballero and Chaumont (2006a,b); Chaumont et al. (2009); Patie (2009, 2012); Bogdan and Żak (2006); Patie (2012); Kyprianou et al. (2014); Kuznetsov et al. (2014); Kyprianou and Watson (2014); Kuznetsov and Pardo (2013); Kyprianou (2016); Kyprianou et al. (2016a,b); Alili et al. (2017). In this review article, we will rediscover many of the aforementioned classical identities in relation to the unit ball by combining elements of these two theories, which have otherwise been largely separated by decades in the literature. We present a dialogue that appeals as much as possible to path decompositions. Most notable in this respect is the Lamperti-Kiu decomposition of self-similar Markov processes given in Kiu (1980); Chaumont et al. (2013); Alili et al. (2017) and the Riesz–Bogdan–Żak transformation given in Bogdan and Żak (2006). Some of the results and proofs we give are known (marked ♥), some are mixed with new results or methods, respectively, (marked ♦) and some are completely new (marked ♣). We assume that the reader has a degree of familiarity with the bare basics of Lévy processes but, nonetheless, we often include reminders of standard material that can be found in e.g. Bertoin (1996), Sato (2013) or Kyprianou (2014). Stable Lévy processes, self-similarity and the unit ball 3
18 citations
Cites background from "Lévy processes and infinitely divis..."
...The subtleties of this can be found for, example, in Chapter 8, Section 43 of Sato (2013)....
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...(3) For a derivation of this exponent, see Exercise 1.4 of Kyprianou (2014) or Chapter 3 of Sato (2013)....
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...We assume that the reader has a degree of familiarity with the bare basics of Lévy processes but, nonetheless, we often include reminders of standard material that can be found in e.g. Bertoin (1996), Sato (2013) or Kyprianou (2014)....
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TL;DR: In this paper, a model of intermittency based on Levy driven Ornstein-Uhlenbeck (OU) type processes is proposed and a discrete superposition of these processes can be constructed to incorporate nonGaussian marginal distributions and long or short range dependence.
Abstract: The phenomenon of intermittency has been widely discussed in physics literature. This paper provides a model of intermittency based on Levy driven Ornstein–Uhlenbeck (OU) type processes. Discrete superpositions of these processes can be constructed to incorporate non-Gaussian marginal distributions and long or short range dependence. While the partial sums of finite superpositions of OU type processes obey the central limit theorem, we show that the partial sums of a large class of infinite long range dependent superpositions are intermittent. We discuss the property of intermittency and behavior of the cumulants for the superpositions of OU type processes.
18 citations
TL;DR: In this paper, the recovery problem of the underlying random clock of the process is considered, and a threshold type estimator based on high-frequency discrete observations is proposed to estimate the random clock, the Blumenthal-Geetor index of jump activity, and the spectral Levy measure.
Abstract: Volatility clustering and leverage are two of the most prominent stylized features of the dynamics of asset prices. In order to incorporate these features as well as the typical fat-tails of the log return distributions, several types of exponential Levy models driven by random clocks have been proposed in the literature. These models constitute a viable alternative to the classical stochastic volatility approach based on SDEs driven by Wiener processes. This paper has two main objectives. First, using threshold type estimators based on high-frequency discrete observations of the process, we consider the recovery problem of the underlying random clock of the process. We show consistency of our estimator in the mean-square sense, extending former results in the literature for more general Levy processes and for irregular sampling schemes. Secondly, we illustrate empirically the estimation of the random clock, the Blumenthal-Geetor index of jump activity, and the spectral Levy measure of the process using real intraday high-frequency data.
18 citations
Cites background from "Lévy processes and infinitely divis..."
...Keywords and phrases: Lévy processes, nonparametric estimation, highfrequency based inference, stochastic volatility, random time-changes....
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TL;DR: In this article, the authors considered escape probability for dynamical systems driven by non-Gaussian Levy motions, especially symmetric α-stable Levy motions and characterized the escape probabilities as solutions of the Balayage-Dirichlet problems of certain partial differential-integral equations.
Abstract: The escape probability is a deterministic concept that quantifies some aspects of stochastic dynamics. This issue has been investigated previously for dynamical systems driven by Gaussian Brownian motions. The present work considers escape probabilities for dynamical systems driven by non-Gaussian Levy motions, especially symmetric α-stable Levy motions. The escape probabilities are characterized as solutions of the Balayage-Dirichlet problems of certain partial differential-integral equations. Differences between escape probabilities for dynamical systems driven by Gaussian and non-Gaussian noises are highlighted. In certain special cases, analytic results for escape probabilities are given.
18 citations
Cites background from "Lévy processes and infinitely divis..."
...Because Lt has càdlàg and quasi-left-continuous paths([24]), Xt(x) also has càdlàg and quasi-left-continuous paths....
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Posted Content•
TL;DR: In this paper, a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift is presented, where the time change is given by a general inverse subordinator.
Abstract: This paper establishes a discretization scheme for a large class of stochastic differential equations driven by a time-changed Brownian motion with drift, where the time change is given by a general inverse subordinator. The scheme involves two types of errors: one generated by application of the Euler-Maruyama scheme and the other ascribed to simulation of the inverse subordinator. With the two errors carefully examined, the orders of strong and weak convergence are derived. Numerical examples are attached to support the convergence results.
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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
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
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
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