Book•
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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Dissertation•
23 Jun 2015
TL;DR: In this paper, the authors studied the properties of d-dimensional jump type diffusion with infinitesimal generator given by Lψ(x) = 1/2 ∑ aᵤᵥ(x)/∂xᵥ∂∂ xᵥ + g(x),∇ψ (x) + ∫ (ψ,x + c(z, x)) − ψ, x)µ(dz) where µ is of infinite total mass, and gave sufficient conditions in order to obtain existence and uniqueness of
Abstract: This work is dedicated to the study of some properties concerning the d-dimensional jump type diffusion X = (Xt) with infinitesimal generator given by Lψ(x) = 1/2 ∑ aᵤᵥ(x)∂²ψ(x)/∂xᵤ∂xᵥ + g(x)∇ψ(x) + ∫ (ψ(x + c(z, x)) − ψ(x))γ(z, x)µ(dz) where µ is of infinite total mass. If γ did not depend on x, we would be in a classical situation where the process X could be represented as the solution of a stochastic equation driven by a Poisson point measure with intensity measure γ(z)µ(dz) ; when γ depends on x, we may have the heuristic idea that, if we were to imagine the process as a trajectory of a particle, the law of the jumps may depend on the position of the particle. In the first part, we give some conditions to obtain existence and uniqueness of such processes. Then, we consider this type of processes as a generalization of Piecewise Deterministic Markov Processes (PDMP) ; we show that they can be seen as a limit of a sequence (Xᵣ(t)) of standard PDMP's for which the intensity of the jumps tends to infinity as r tends to infinity, following two regimes: a slow one, which leads to a jump component with finite variation, and a rapid one which, supposing that the processes at hand are centered and renormalized in a convenient way, produces the diffusion component in the limit. Finally, we prove Harris recurrence of X using a regeneration scheme which is entirely based on the jumps of the process. Moreover we state explicit conditions in terms of the coefficients of the process allowing to control the speed of convergence to equilibrium in terms of deviation inequalities for integrable additive functionals. In the second part, we consider again the same type of process X = (Xt(x)) starting from x. Using an approach based on a finite dimensional Malliavin Calculus, we study the joint regularity of this process in the following sense : we fix b≥1 and p>1, K a compact set of Rᵈ, and we give sufficient conditions in order to have P(Xt(x)∈dy)=pt(x,y)dy with (x,y)↦pt(x,y) in Wᵇᵖ(K×Rᵈ)
6 citations
Posted Content•
TL;DR: In this paper, the authors consider a nonlinear random walk which is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Levy process.
Abstract: We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Levy process. In analogy to the classical framework we show that, when passing from discrete to continuous time via a scaling limit, this nonlinear random walk gives rise to a nonlinear semigroup. We explicitly compute the generator of this semigroup and corresponding PDE as a perturbation of the generator of the initial Levy process.
6 citations
TL;DR: In this article, a tractable market impact model for electricity futures prices is proposed and optimal liquidation strategies with respect to different target functionals of "conditional expected trading cost" type are derived.
Abstract: Electricity markets commonly exhibit an oligopolistic structure with market participants whose individual trading activities may shift prices essentially. In this context, the question of how to optimally liquidate an existing electricity futures portfolio over a fixed time horizon under the constraint of minimizing unfavorable price impact effects is of striking relevance for energy risk management. In this article, we thus invent a tractable market impact model for electricity futures prices. More precisely, we derive optimal liquidation strategies with respect to different target functionals of ‘conditional expected trading cost’ type. Moreover, we take forward-looking information about future electricity price behavior into account by exploiting enlarged filtration methods. Finally, we derive optimal liquidation strategies for electricity futures portfolios under this insider trading machinery as well.
6 citations
Cites background from "Lévy processes and infinitely divis..."
...[6], [11], [23], [26], [27]) which leads us to the F -adapted representation...
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TL;DR: Under minimal assumptions it is proved that convergence of the optimal discrete-time strategies to the continuous-time counterpart is convergence.
Abstract: Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.
6 citations
TL;DR: A quantitative criterion for choice of the sampling rate at which a spatial model with independent components resembles a rotation-invariant model is proposed, which has the potential to assist the observer to employ simpler models in the continuous-time framework to avoid expensive computation required for statistical inference.
Abstract: Continuous-time modeling of random searches is designed to be robust to the sampling rate while the spatial model is required to be of rotation-invariant type, which is often computationally prohibitive. Such computational difficulty may be circumvented by employing a model with independent components. We demonstrate that its disadvantages in statistical properties are blurred under lower frequency. We propose a quantitative criterion for choice of the sampling rate at which a spatial model with independent components resembles a rotation-invariant model. Our findings have the potential to assist the observer to employ simpler models in the continuous-time framework to avoid expensive computation required for statistical inference.
6 citations
Cites background or methods from "Lévy processes and infinitely divis..."
...As the univariate stable motion is thoroughly studied [11, 12] and the multivariate motion is still tractable in some instances, our discussion does not require unnecessarily intricate derivations but the basic known facts....
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...In this article, we focus on a (nonGaussian) two-dimensional symmetric stable (Lévy) motion {Xt : t ≥ 0}, which can be defined through the characteristic function [11, 12]...
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...It is well known that the probability tails behave like [11, 12]...
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...Each component has the so-called inverse power-law tail; a random variable Z with E[eiyZ ] = exp[−σα |y|α ] has the probability tail behavior [11, 12]...
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...This stable motion, with γ = 0, that is, no bias in movements, enjoys the self-similarity [11, 12, 13, 14, 15]; for any h > 0, { h−1/α Xht : t ≥ 0 } L = {Xt : t ≥ 0} , (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