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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: A favorable maximum likelihood parameter estimation scheme for the discrete distribution is introduced based on a quadrature method by approximating the special function by the trapezoidal rule and the effectiveness of the method is confirmed by expectation–maximization (EM) algorithm.
Abstract: A generalization of the geometric distribution is obtained after mixing the Poisson distribution with the generalized exponential distribution in Marshall and Olkin in 1997. The discrete distribution is defined through a new special function also introduced in this manuscript. Unimodality is highlighted among the properties of this two-parameter distribution. A favorable maximum likelihood parameter estimation scheme for the discrete distribution is introduced based on a quadrature method by approximating the special function by the trapezoidal rule. The effectiveness of the method is confirmed by expectation–maximization (EM) algorithm. An application to explain the demand for health services is given.
8 citations
TL;DR: In this paper, Misiewicz et al. define a weak generalized convolution of measures defined by the formula, which is equivalent to the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that ${\bf X} Q_1 + q_2' Q-2 \stackrel{d}(Q_2) = {\cal L}(\Theta),$ if the former equation holds for all variables.
Abstract: A random vector ${\bf X}$ is weakly stable if and only if for all $a,b \in {\mathbb R}$ there exists a random variable $\Theta$ such that $a{\bf X} + b {\bf X}' \stackrel{d}{=} {\bf X} \Theta$, where ${\bf X}'$ is an independent copy of ${\bf X}$ and $\Theta$ is independent of ${\bf X}$. This is equivalent (see [J. K. Misiewicz, K. Oleszkiewicz, and K. Urbanik, Studia Math., 167 (2005), pp. 195--213]) to the condition that for all random variables $Q_1, Q_2$ there exists a random variable $\Theta$ such that ${\bf X} Q_1 + {\bf X}' Q_2 \stackrel{d}{=} {\bf X} \Theta,$ where ${\bf X}, {\bf X}', Q_1, Q_2, \Theta$ are independent. In this paper we define a weak generalized convolution of measures defined by the formula ${\cal L}(Q_1) \otimes_{\mu} {\cal L}(Q_2) = {\cal L}(\Theta),$ if the former equation holds for ${\bf X}, Q_1, Q_2, \Theta$ and $\mu = {\cal L}(\bf X)$. We study here basic properties of this convolution and basic properties of distributions which are infinitely divisible in the sense of this ...
8 citations
Cites background or methods from "Lévy processes and infinitely divis..."
...This replacement allows us to use weak convergence technics for Lévy spectral measures because of continuity of functions c ∈ C. Sato (see [17])showed that the Lévy measure can be obtained as a weak limit (in somewhat restricted sense) of a sequence of measures defined by convolution powers of the considered infinitely divisible distribution....
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...In this section we will use the construction of the Lévy measure for infinitely divisible distribution given in the book of Sato [17], section 8 in order to show that µ-infinitely divisible mixture of weakly stable measure is also a mixture of this measure....
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...Since λ is μ-weakly infinitely divisible then η = μ ◦ λ is infinitely divisible in the usual sense and according to the proof of Theorem 8(i) in [17] we have Exp ( t n η ∗tn ) → η, n→ ∞....
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...In this section we will use the construction of the Lévy measure for infinitely divisible distribution given in the book of Sato [17], section 8 in order to show that μ-infinitely divisible mixture of weakly stable measure is also a mixture of this measure....
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...Sato (see [17])showed that the Lévy measure can be obtained as a weak limit (in somewhat restricted sense) of a sequence of measures defined by convolution powers of the considered infinitely divisible distribution....
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Posted Content•
TL;DR: In this article, an upper bound for the Wasserstein distance in the general framework of the Wiener-Poisson space is provided, and a second order Poincare-type inequality is obtained from this bound.
Abstract: An upper bound for the Wasserstein distance is provided in the general framework of the Wiener-Poisson space. Is obtained from this bound a second order Poincare-type inequality which is useful in terms of computations. For completeness sake, is made a survey of these results on the Wiener space, the Poisson space, and the Wiener-Poisson space, and showed several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated field (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein-Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to \small" jumps (particularly fractional Levy processes) and the product of two Ornstein-Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). Also, are obtained bounds for their rate of convergence to normality.
8 citations
Cites background from "Lévy processes and infinitely divis..."
...[33] K. Sato (1999)....
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...A fuller exposition on Lévy processes can be found in [2] and [33]....
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01 Jan 2012
TL;DR: In this paper, the Malliavin derivative operator for a Lévy process (Xt) was defined on the space D1,2 using a chaos expansion or in the case of a pure jump process also via an increment quotient operator.
Abstract: The Malliavin derivative for a Lévy process (Xt) can be defined on the space D1,2 using a chaos expansion or in the case of a pure jump process also via an increment quotient operator. In this paper we define the Malliavin derivative operator D on the class S of smooth random variables f(Xt1 , . . . , Xtn ), where f is a smooth function with compact support. We show that the closure of L2(P) ⊇ S D → L2(m⊗P) yields to the space D1,2. As an application we conclude that Lipschitz functions operate on D1,2. 2000 AMS Mathematics Subject Classification: Primary: 60H07; Secondary: 60G51.
8 citations
Cites background from "Lévy processes and infinitely divis..."
...[36] and Davis and Johansson [16] as well as Benth et al....
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...This chaotic representation was shown by Nualart and Schoutens [43], and Malliavin calculus based on it has been studied by authors such as Leon et al. [36] and Davis and Johansson [16] as well as Benth et al. [7] and Solé et al. [57], who compare the two chaos expansion based approaches....
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...[16] M. Davis and M. Johansson....
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01 Jan 2011
TL;DR: Carmona et al. as mentioned in this paper introduce an arbitrage-free Markt model for stock options based on the Levy-Dichte Codebuch and a Markt-Martingal model.
Abstract: Gebrauchliche Aktienpreis- bzw.Martingalmodelle beschreiben die Dynamik des Preises einer Aktie unter einem Martingalmas; grundlegendes Beispiel ist das Black-Scholes Modell. Im Gegensatz dazu zielt ein Marktmodell darauf ab die Dynamik eines ganzen Marktes (d.h. Aktienpreis plus abgeleitete Optionen) zu beschreiben. In der vorliegenden Arbeit
beschaftigen wir uns mit arbitragefreien Modellen die die Dynamik eines Aktienpreises sowie flussig gehandelter Derivate beschreiben. ("equity market models" respektive "market models for stock options").
Die Motivation derartige Modelle zu betrachten, liegt darin, dass europaische Optionen flussig gehandelt werden und daher Ruckschlusse auf die zugrundeliegende Stochastik erlauben.
Marktmodelle werden auch fur Anleihen angewandt; wir verweisen auf Heath, Jarrow, and Morton [1992]. In jungerer Zeit beeinflust dieser Zugang auch Aktienpreismodelle (siehe beispielsweise Derman and Kani [1998], Dupire [1996]). Derman und Kani schlagen vor, die dynamische
Entwicklung von Markten mittels Differentialgleichungen fur Aktienpreis und volatility surface zu modellieren.
Ein anderer Zugang wird von Schonbucher [1999] gewahlt; hier ist der Ausgangspunkt die gemeinsame Dynamik von Aktienpreis und impliziten Black-Scholes Volatilitaten. Carmona and Nadtochiy [2009] schlagen ein Marktmodell vor, in dem der Aktienpreis als exponentieller Levy-Prozes gegeben ist. In diesem Fall wird eine zeitinhomogene Levy-Dichte verwendet
um die zusatzlichen, durch Optionspreise gegebenen Informationen miteinzubeziehen.
Der erste Teil dieser Dissertation beschreibt zwei unterschiedliche Zugange um durch den Markt gegebene Informationen zu beschreiben. Einerseits kann dies durch ein local volatility Codebuch geschehen, andererseits kann ein Levy-Dichte Codebuch verwendet werden. Wir beschreiben verschiedene Kalibrierungsverfahren; insbesondere vergleichen wir parametrische mit nicht-parametrischen Methoden.
Im zweiten Abschnitt beschreiben wir verschiedene dynamische Modelle, die entstehen, wenn man die beiden obigen Codebucher "in Bewegung setzt". D.h. wir beschreiben dynamische local-volatility sowie dynamische Levy-Dichte Modelle. Besonderes Augenmerk liegt dabei auf der Konsistenz
(d.h. Aribtragefreiheit) dieser Modelle. Aufgrund der zusatzlichen stochastischen Dynamik, erscheint dieser Aspekt durchaus als Herausforderung.
Im letzten Teil dieser Arbeit betrachten wir einen speziellen Typ von Levy-Modellen. Die spezielle Struktur der Modellparameter ist durch eine Kalibrierung an Marktdaten gemas Ortega et al. [2009] motiviert. Wir berechnen Konsistenzbedingungen fur den vorliegenden speziellen Typ
und implementieren sie mithilfe eines geeigneten Euler-Schemas.
8 citations
Cites result from "Lévy processes and infinitely divis..."
...Schmitz [2004] show that calculating the local volatility using the above formula gives a more accurate and stable result compared with the formula (2....
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References
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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