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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 paper, the authors considered the Dirichlet problems associated with integro-differential elliptic operators of order α < 1 in a bounded smooth domain and concluded strong comparison principles in under rather general assumptions over the drift term.
Abstract: In this paper we are interested on the well-posedness of Dirichlet problems associated to integro-differential elliptic operators of order α <1 in a bounded smooth domain Ω. The main difficulty arises because of losses of the boundary condition for sub and supersolutions due to the lower diffusive effect of the elliptic operator compared with the drift term. We consider the notion of viscosity solution with generalized boundary conditions, concluding strong comparison principles in under rather general assumptions over the drift term. As a consequence, existence and uniqueness of solutions in is obtained via Perron's method.
13 citations
Cites background from "Lévy processes and infinitely divis..."
...When − = − /2, it is known that (see [23]) the fractional Laplacian is the infinitesimal...
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TL;DR: In this paper, the authors characterize the small-time asymptotic behavior of the exit probability of a Levy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process.
Abstract: We characterize the small-time asymptotic behavior of the exit probability of a Levy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed Levy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural credit risk models, pricing of barrier options, and contingent convertible bonds) as well as in natural sciences. The proposed algorithm works by adding discretization points sampled from the Levy bridge density to the skeleton of the process until the overall error for a given trajectory becomes smaller than the maximum tolerance given by the user.
13 citations
TL;DR: Spectral theory for transition operators of one-dimensional symmetric Levy process killed upon hitting the origin is studied in this paper, where integral fomulae for the transition density and the distribution of the hitting time of the origin are given in terms of the eigenfunctions.
Abstract: Spectral theory for transition operators of one-dimensional symmetric Levy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction expansion of transition operators and the generator is proved. As an application, and the primary motivation, integral fomulae for the transition density and the distribution of the hitting time of the origin are given in terms of the eigenfunctions.
13 citations
Cites background from "Lévy processes and infinitely divis..."
...3 in [16], 0 is regular for itself, and so P t is a strongly continuous semigroup of operators on C0(R \ 0) (see also [12])....
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...3 in [16], uz(x)/uz(0) is the Laplace transform of the distribution of τ0, Exe −zτ0 = uz(x) uz(0) , z > 0, x ∈ R....
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...Some more detailed results in this area have been obtained for spectrally negative Lévy processes, see, for example, [6] and Section 46 in [16]....
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...General references for Lévy processes are [3, 16]; for the properties of semigroups of killed Lévy processes, see [12] and the references therein....
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TL;DR: An axiomatic framework for decision making based directly on historical time series is presented and used for constructing data-based analogues of mean-variance and maxmin utility approaches to optimal portfolio selection.
13 citations
Cites background or methods from "Lévy processes and infinitely divis..."
...Stochastic processes with stationary and independent increments (and with discontinuous sample paths in general), are called Lévy processes [48] and nowadays are widely-used in modeling of financial markets [31, 37, 41]....
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...Stochastic processes with stationary and independent increments (and with discontinuous sample paths in general), are called Lévy processes ( Sato, 1999 ) and nowadays are widely-used in modeling of financial markets ( Kou, 2002; Madan & Seneta, 1990; Merton, 1976 )....
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...For example, autoregressive models AR( p ) 1 Sato processes ( Sato, 1991 ), whose increments are independent but not neces- sarily stationary, can be used instead. https://doi.org/10.1016/j.ejor.2017.11.021 0377-2217/© 2017 Elsevier B.V....
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...Black & Scholes [5] priced options based on the assumption that stock prices follow a Brownian motion, whereas Sato [48] replaced the Brownian motion by a Levy process, which also assumes price increments (and consequently returns) to be independent....
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TL;DR: In this article, three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field were studied and the Euler characteristic estimator was used to determine whether a given field is Gaussian or not, when compared to various alternatives.
Abstract: In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.
13 citations
Cites background from "Lévy processes and infinitely divis..."
...In the case where B is uniformly distributed in {−1,+1}, the field SΦ is symmetric and its marginals have a Skellam distribution with both parameters equal to νā/2 (see [30]), that is to say that they coincide with the difference of two independent Poisson random variables with parameter νā/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