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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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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
Cites background from "Lévy processes and infinitely divis..."
...There can be no doubt, particularly to the more experienced reader, that the current text has been heavily influenced by the outstanding books of Bertoin (1996) and Sato (1999), and especially the former which also takes a predominantly pathwise approach to its content....
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...See Zolotarev (1986), Sato (1999) and (Samorodnitsky and Taqqu, 1994) for further details of all the facts given in this paragraph....
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...The interested reader is referred to Lukacs (1970) or Sato (1999), to name but two of many possible references....
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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
Cites background from "Lévy processes and infinitely divis..."
...Distributional definition of L is also studied in [7, 34, 42], see also [3, 43]....
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...Yet another way to show (10) involves vague convergence of tpt(z)dz to ν(z)dz = cd,α|z|dz as t → 0, which is a general result in the theory of convolution semigroups, see [43]....
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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
TL;DR: For the isotropic unimodal probability convolutional semigroups, this article gave sharp bounds for their Levy-Khintchine exponent with Matuszewska indices strictly between 0 and 2.
172 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
References
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TL;DR: In this article, the slow dynamics of a nonlocal fast-slow stochastic evolutionary system with stable Levy noise were investigated. And two examples with numerical simulations were presented to illustrate the results.
Abstract: This work aims at understanding the slow dynamics of a nonlocal fast-slow stochastic evolutionary system with stable Levy noise. Slow manifolds along with the exponential tracking property for a nonlocal fast-slow stochastic evolutionary system with stable Levy noise are constructed, and two examples with numerical simulations are presented to illustrate the results.
6 citations
TL;DR: In this article, the authors measured market efficiency by arbitrage proximity and the magnitude of probability distortion necessary to remove drift calibrates the efficiency, and the relative inefficiency of the absence of trading is noted on comparing close to open return efficiencies those for open to close returns.
Abstract: Market efficiency is measured by arbitrage proximity. The magnitude of probability distortion necessary to remove drift calibrates the efficiency. Simulations of bilateral gamma models estimated on a year's past returns yield empirical acceptability indices for each day for each asset. The assets covered include, equities, commodities, currencies and volatility. It is observed that efficiency in equity is related to the up side gamma process having more frequent and smaller jumps than its down side counterparts. For commodities the situation is reversed. The relative inefficiency of the absence of trading is noted on comparing close to open return efficiencies those for open to close returns. Small capitalization stocks trade more efficiently than large capitalization stocks. Sector exchange traded funds trade more efficiently than the S&P 500 index. Furthermore, economic activity reflected in greater high lows spreads enhances market efficiency.
6 citations
Posted Content•
TL;DR: In this article, a non-homogeneous Poisson cluster model is studied, motivated by insurance applications, where the Poisson center process which expresses arrival times of claims, triggers off cluster member processes which correspond to number or amount of payments.
Abstract: A non-homogeneous Poisson cluster model is studied, motivated by insurance applications. The Poisson center process which expresses arrival times of claims, triggers off cluster member processes which correspond to number or amount of payments. The cluster member process is an additive process. Given the past observations of the process we consider expected values of future increments and their mean squared errors, aiming the application in claims reserving problems. Our proposed process can cope with non-homogeneous observations such as the seasonality of claims arrival or the reducing property of payment processes, which are unavailable in the former models where both center and member processes are time homogeneous. Hence results presented in this paper are significant extensions toward applications. We also give numerical examples to show how non-homogeneity appears in predictions.
6 citations
Dissertation•
16 Oct 2015
TL;DR: In this paper, the authors consider the problem of estimating the density of the law of the solution at a fixed, short time, and then use these short-time estimates to show exponential two-sided bounds for the probability that the diffusion remains in a small tube around a deterministic path up to a given time.
Abstract: In this thesis we address two problems. In the first part we consider hypoelliptic diffusions, under both strong and weak Hormander condition. We find Gaussian estimates for the density of the law of the solution at a fixed, short time. A main tool to prove these estimates is Malliavin Calculus, in particular some techniques recently developed to deal with degenerate problems. We then use these short-time estimates to show exponential two-sided bounds for the probability that the diffusion remains in a small tube around a deterministic path up to a given time. In our hypoelliptic framework, the shape of the tube must reflect the fact the diffusion moves with a different speed in the direction of the diffusion coefficient and in the direction of the Lie brackets. For this reason we introduce a norm accounting of this anisotropic behavior, which can be adapted to both the strong and weak Hormander framework. We establish a connection between this norm and the standard control distance in the strong Hormander case. In the weak Hormander case, we introduce a suitable equivalent control distance. In the second part of the thesis we work with mean reverting stochastic volatility models, with a volatility driven by a jump process. We first suppose that the jumps follow a Poisson process, and consider the decay of cross asset correlations, both theoretically and empirically. This leads us to study an algorithm for the detection of jumps in the volatility profile. We then consider a more subtle phenomenon widely observed in financial indices: the multiscaling of moments, i.e. the fact that the q-moment of the log-increment of the price on a time lag of length h scales as h to a certain power of q, which is non-linear in q. We work with models where the volatility follows a mean reverting SDE driven by a Levy subordinator. We show that multiscaling occurs if the characteristic measure of the Levy has power law tails and the mean reversion is super-linear at infinity. In this case the scaling function is piecewise linear
6 citations
TL;DR: In this article, a stochastic SIS model with Gaussian and Poissonian perturbations was proposed to account for noise and anomalies in the transmission rate, and conditions were given for stability to the disease free equilibrium and for positive Harris recurrence with a unique invariant measure for the endemic.
Abstract: We propose a stochastic SIS model to include both a Gaussian and Poissonian perturbation to account for noise and anomalies in the transmission rate. Conditions are given for stability to the disease free equilibrium and for positive Harris recurrence with a unique invariant measure for the endemic.
6 citations