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

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Fluctuations of Lévy Processes with Applications

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener-Hopf factorization for Levy processes where previously there had been very few. We mention in particular the many cases of spectrally negative Levy processes, hyper-exponential and generalized hyper-exponential Levy processes, Lamperti-stable processes, Hypergeometric processes, Beta-processes and Theta-processes. In this paper we introduce a new family of Levy processes, which we call Meromorphic Levy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener-Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Levy-Khintchin exponent, all of which appear on the imaginary axis of the complex plane. The specific structure of the M-class Wiener-Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

Meromorphic Levy processes and their fluctuation identities

We develop a completely new and straightforward method for simulating the joint law of the position and running maximum at a fixed time of a general Levy process with a view to application in insurance and financial mathematics. Although different, our method takes lessons from Carr’s so-called “Canadization” technique as well as Doney’s method of stochastic bounds for Levy processes; see Carr [Rev. Fin. Studies 11 (1998) 597–626] and Doney [Ann. Probab. 32 (2004) 1545–1552]. We rely fundamentally on the Wiener–Hopf decomposition for Levy processes as well as taking advantage of recent developments in factorization techniques of the latter theory due to Vigon [Simplifiez vos Levy en titillant la factorization de Wiener–Hopf (2002) Laboratoire de Mathematiques de L’INSA de Rouen] and Kuznetsov [Ann. Appl. Probab. 20 (2010) 1801–1830]. We illustrate our Wiener–Hopf Monte Carlo method on a number of different processes, including a new family of Levy processes called hypergeometric Levy processes. Moreover, we illustrate the robustness of working with a Wiener–Hopf decomposition with two extensions. The first extension shows that if one can successfully simulate for a given Levy processes then one can successfully simulate for any independent sum of the latter process and a compound Poisson process. The second extension illustrates how one may produce a straightforward approximation for simulating the two-sided exit problem.

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A Wiener–Hopf Monte Carlo simulation technique for Lévy processes

A WienerHopf Monte Carlo simulation technique for Lévy processes.

We derive explicitly the coupling property for the transition semigroup of a Levy process and gradient estimates for the associated semigroup of transition operators This is based on the asymptotic behaviour of the symbol or the characteristic exponent near zero and infinity, respectively Our results can be applied to a large class of Levy processes, including stable Levy processes, layered stable processes, tempered stable processes and relativistic stable processes

Coupling property and gradient estimates of L\'{e}vy processes via the symbol

Inspired by works of Landriault et al. [11, 12], we study the Gerber{Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Levy insurance risk process. To be more specific, we study the so-called Gerber{Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Levy processes and relies on the theory of so-called scale functions. In particular, we extend recent results of Landriault et al. [11, 12].

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Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

We consider a new family of IR d -valued L evy processes that we call Lamperti stable. One of the advantages of this class is that the law of many related functionals can be computed explicitly. Here, in the one dimensional case we provide an explicit form for the characteristic exponent and other several useful properties of the class (for some particular cases see [6], [10], [24] and [28]). This family of processes shares many properties with the tempered stable and the layered stable processes, dened in Rosi nski [31] and Houdr e and Kawai [16] respectively, for instance their short and long time behavior. We also nd a series representation which is used for sample path simulation, illustrated in the case d = 1. Finally we provide many examples, some of which appear in recent literature.

On the Lamperti stable processes

The coastline of both sides of Mexico exceeds 11 000 km, consisting of various landforms that are regio- nally different in origin, length, and nature, called by the common term of coastal lagoons by different authors and government institutions, and currently called wetlands by international convention (RAMSAR). This study thus differentiates lagoons, estuaries, marshes and bays in the Pacific, Gulf of Mexico and Caribbean. In the Gulf of Mexico predominated the coastal lagoons (47.8%) and the last were the bays (1.6%); and in the Pacific the bays and costal lagoons had the same number of this geoforms (14%), approximately.

http://www.scielo.org.mx/pdf/igeo/n81/n81a4.pdf

Diferenciación hidrogeomorfológica de los ambientes costeros del Pacífico, del Golfo de México y del Mar Caribe

In this paper we introduce a new class of Levy processes which we call hypergeometric-stable Levy processes, because they are obtained from symmetric stable processes through several transformations and where the Gauss hypergeometric function plays an essential role. We characterize the Levy measure of this class and obtain several useful properties such as the Wiener Hopf factorization, the characteristic exponent and some associated exit problems.

Explicit identities for L\'evy processes associated to symmetric stable processes

Consider any supercritical Galton-Watson process which may become extinct with positive probability. It is a well-understood and intuitively obvious phenomenon that, on the survival set, the process may be pathwise decomposed into a stochastically ‘thinner’ Galton-Watson process, which almost surely survives and which is decorated with immigrants, at every time step, initiating independent copies of the original Galton-Watson process conditioned to become extinct. The thinner process is known as the backbone and characterizes the genealogical lines of descent of prolific individuals in the original process. Here, prolific means individuals who have at least one descendant in every subsequent generation to their ownn.

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José Luis Carbajal Pérez

Papers

Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

On the Lamperti stable processes

Diferenciación hidrogeomorfológica de los ambientes costeros del Pacífico, del Golfo de México y del Mar Caribe

Explicit identities for L\'evy processes associated to symmetric stable processes

The Backbone Decomposition for Spatially Dependent Supercritical Superprocesses