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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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Posted Content
TL;DR: In this article, the authors considered multidimensional polynomial Euler products with complex coefficients and gave necessary and sufficient conditions for them to generate infinitely divisible, quasi-infinitely divisible but not even characteristic functions by using Baker's theorem.
Abstract: In the present paper, we treat multidimensional polynomial Euler products with complex coefficients on ${\mathbb{R}}^d$. We give necessary and sufficient conditions for the multidimensional polynomial Euler products to generate infinitely divisible, quasi-infinitely divisible but non-infinitely divisible or not even characteristic functions by using Baker's theorem. Moreover, we give many examples of zeta distributions on ${\mathbb{R}}^d$ generated by the multidimensional polynomial Euler products with complex coefficients. Finally, we consider applications to analytic number theory.

6 citations

Journal ArticleDOI
TL;DR: The paper addresses the valuation of contingent claims in stochastic volatility models of Ornstein-Uhlenbeck type, stressing the situation when volatility is driven by purely-discontinuous Levy processes, and develops a reduction series methodology for this purpose.

6 citations

Dissertation
04 Sep 2019

6 citations


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

  • ...For an introduction to more general Markov processes, see, [55], [72], [92]....

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  • ...The following result is well-known of general Dirichlet forms and generators of Lévy processes, see [231], [92] for more general results in this direction....

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  • ...Hence, TD(t) form a contraction semigroup on each of the spaces L(D) (p ∈ [1,∞]), and if ψ is unbounded, then also on C0(D) (see [228], [112], [92])....

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  • ...We consider the pure-jump L?́?vy process X = (Xt , t ≥ 0) on R d [92], in short: Xt , determined by the L?́?vy-Khintchine formula E eiξ,Xt = e−tψξ = ∫ ept(dx) Rd ....

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Posted Content
TL;DR: For the multivariate COGARCH(1,1) volatility process, this paper showed sufficient conditions for the existence of a unique stationary distribution, for the geometric ergodicity and for the finiteness of moments of the stationary distribution.
Abstract: For the multivariate COGARCH(1,1) volatility process we show sufficient conditions for the existence of a unique stationary distribution, for the geometric ergodicity and for the finiteness of moments of the stationary distribution. One of the conditions demands a sufficiently fast exponential decay of the MUCOGARCH(1,1) volatility process. Furthermore, we show easily applicable sufficient conditions for the needed irreducibility of the volatility process living in the cone of positive semidefinite matrices, if the driving L\'evy process is a compound Poisson process.

6 citations

DissertationDOI
08 Sep 2011
TL;DR: In this paper, the authors studied the generalization of reaction-subdiffusion schemes to subdiffusion by means of continuous-time random walks on a mesoscopic scale with a heavy-tailed waiting time pdf ψ(t) ∝ t−1−α lacking the first moment.
Abstract: The present work studies the generalization of reaction–diffusion schemes to subdiffusion. The subdiffusive dynamics was modelled by means of continuous–time random walks on a mesoscopic scale with a heavy–tailed waiting time pdf ψ(t) ∝ t−1−α lacking the first moment. The reaction itself was assumed to take place on a microscopic scale, obeying the classical mass action law. This situation is assumed to apply in a porous medium where the particles are trapped within the catchments, pores and stagnant regions of the flow, but are still able to react during their waiting times. After discussing the subdiffusion equation and different methods of their solution, especially under the aspect of particles being introduced into the system in the course of time, the reaction–subdiffusion equations are addressed. These equations are of integro–differential form and under the assumptions made, the reaction explicitly affects the transport term. The long ranged memory of the subdiffusion kernel is modified by an additional factor accounting for the conversion and survival probabilities due to reaction during the waiting times. In the case of linear reaction kinetics, this factor is governed by the rate coefficients. For nonlinear reaction kinetics the transport kernel depends additionally on the concentrations of the respective reaction partners at all previous times. The simplest linear reaction, the degradation A→ 0 was considered and a general expression for the solution to arbitrary Dirichlet Boundary Value Problems was derived. This solution can be expressed in terms of the solution to the corresponding Dirichlet Problem under mere subdiffusion, i.e. without degradation. The resultant stationary profiles do not differ qualitatively from the stationary profiles in normal reaction diffusion. For stationary solutions to exist in reaction–subdiffusion, the assumption of reactions according to classical rate kinetics is essential. As an example for a nonlinear reaction–subdiffusion system, the irreversible autocatalytic reaction A + B→ 2A under subdiffusion is considered. Under the assumptions of constant overall particle concentration A(x, t) + B(x, t) = const and re–labelling of the converted particles, a subdiffusive analogue of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) equation was derived and the resultant fronts of A–particles propagating into the B– domain were studied. Two different regimes were detected in numerical simulations. These regimes were discussed using both crossover arguments and analytic calculations. The first regime can be described within the framework of the continuous reaction–subdiffusion equations and is characterized by the front velocity and width going as t α−1 2 at larger times. As the front width decays, the front gets atomically sharp at very large times and a transition to a second regime, the fluctuation dominated one, is expected. The fluctuation dominated regime is not within the scope of the continuous description. In that case, the velocity of the front decays faster in time than in the continuous regime, v f luct ∝ tα−1. Further simulations pertaining the reaction on contact scenario, i.e. the fluctuation dominated regime, revealed additional fluctuation effects that are genuinely due to subdiffusion. Another nonlinear reaction–subdiffusion system where reactants A penetrate a medium initially filled with immobile reactants B and react according to the scheme A+B→ (inert) was considered. Under certain presumptions, this problem can be described in terms of a moving boundary problem, a so–called Stefan–problem, for the concentration of a single species. The main result was that the propagation of the moving boundary between the A– and B–domain goes as R(t) ∝ tα/2. The theoretical predictions concerning the moving boundary were corroborated by numerical simulations.

6 citations

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