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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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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


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....

    [...]

  • ...The interested reader is referred to Lukacs (1970) or Sato (1999), to name but two of many possible references....

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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


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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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

Journal ArticleDOI
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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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

Posted Content
TL;DR: In this article, the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Levy noise was studied and the existence of stable and unstable foliations for this system via the Lyapunov-Perron method was established.
Abstract: This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Levy noise. We first establish the existence of stable and unstable foliations for this system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example.

6 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the convergence rate of the weak Euler approximation for solutions to Levy-driven stochastic differential equations with non-egenerate main part driven by a spherically symmetrized sparsification.
Abstract: This paper studies the rate of convergence of the weak Euler approximation for solutions to Levy-driven stochastic differential equations with nondegenerate main part driven by a spherically symmet...

6 citations

Journal ArticleDOI
TL;DR: In this paper, a non-local nonlinear partial integro-differential equation (PIDE) solution to the problem of pricing call and put options under a Levy stochastic process with jumps is presented.
Abstract: In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Levy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black–Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Levy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Levy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Levy stochastic processes. We also present numerical results and comparison of option prices for various Levy stochastic processes modelling underlying asset dynamics.

6 citations

Dissertation
29 Jun 2012
TL;DR: In this article, the authors study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold.
Abstract: In this thesis I'm interested in two aspects of portfolio management: the portfolio insurance under a risk measure constraint and quadratic hedge in incomplete markets. Part I. I study the problem of portfolio insurance from the point of view of a fund manager, who guarantees to the investor that the portfolio value at maturity will be above a fixed threshold. If, at maturity, the portfolio value is below the guaranteed level, a third party will refund the investor up to the guarantee. In exchange for this protection, for which the investor pays a given fee, the third party imposes a limit on the risk exposure of the fund manager, in the form of a convex monetary risk measure. The fund manager therefore tries to maximize the investor's utility function subject to the risk measure constraint. I give a full solution to this nonconvex optimization problem in the complete market setting and show in particular that the choice of the risk measure is crucial for the optimal portfolio to exist. An interesting outcome is that the fund manager's maximization problem may not admit an optimal solution for all convex risk measures, which means that not all convex risk measures may be used to limit fund's exposure in this way. I provide conditions for the existence of the solution. Explicit results are provided for the entropic risk measure (for which the optimal portfolio always exists), for the class of spectral risk measures (for which the optimal portfolio may fail to exist in some cases) and for the G-divergence. Key words: Portfolio Insurance; utility maximization; convex risk measure; VaR, CVaR and spectral risk measure; entropy and G-divergence. Part II. In the second part I study the problem of quadratic hedge in incomplete markets. I work with a three-dimensional Markov jump process: the first component is the state variable representing the hedging instrument traded in the market, the second component models a risk factor which "perturbs" the dynamics of the hedging instrument and is not traded in the market (as a volatility factor for example in stochastic volatility models); the third one is another source of risk which affects the option's payoff at maturity and is also not traded in the market. The problem can be seen then as a constrained quadratic hedge problem. I privilege here the dynamic programming approach which allows me to obtain the HJB equation related to the value function. This equation is semi linear and non local due the presence of jumps. The main result of this thesis is that this value function, as a function of the initial wealth, is a second order polynomial whose coefficients are characterized as the unique smooth solutions of a triplet of PIDEs, the first of which is semi linear and does not depend on the particular choice of option one wants to hedge, the other two being simply linear. This result is stated when the Markov process is assumed to be a non-generate jump-diffusion and when it is a pure jump process. I finally apply my theoretical results to an example of quadratic hedge in the context of electricity markets.

6 citations