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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 stronger properties of the Toeplitz matrix and Hankel matrix of the generalized cycle index polynomials, and gave some criteria for total positivity of these lower-triangles.
Abstract: Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J Combin Theory Ser A 74 (1996)] Schirmacher [J Combin Theory Ser A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix
By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomials sequence in terms of the exponential formula, the logarithmic formula and the fractional formula
Finally, we apply our criteria to some triangular arrays satisfying some recurrence relations, including Bessel triangles of two kinds and their generalizations, the Lah triangle and its generalization, the idempotent triangle and some triangles related to binomial coefficients We not only get total positivity of these lower-triangles, and $q$-Stieltjes moment properties and $3$-$q$-log-convexity of their row-generating functions, but also prove that their triangular convolutions preserve Stieltjes moment property
6 citations
Cites background from "Lévy processes and infinitely divis..."
...Its log-concavity and log-convexity are very significant in probability and statistics since they play a crucial role in the class of infinitely divisible distributions, one of most important probability distributions in both theory and applications, see [22, 23, 45]....
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01 Jan 2019
TL;DR: In this paper, the authors compare the precios of different models for opciones de barrera with the corresponding prices of the Bolsa alemana, a stock market index of the Eurozone.
Abstract: ##English Version below##
Esta tesis consta de tres ensayos. En el primer ensayo, probamos empiricamente el desempeno de los precios de varios modelos financieros avanzados para opciones exoticas. Calibramos seis modelos avanzados para precios de acciones a una serie de datos de mercado reales de opciones europeas en el DAX, el indice de referencia de la Bolsa alemana. A traves de una simulacion de Monte Carlo, calculamos precios de opciones de barrera para todos los modelos y comparamos los precios modelados con los precios del mercado de las opciones de barrera. El modelo Bates reproduce bien los precios de las opciones de barrera. El modelo BNS sobrevalora y los modelos Levy con cambio temporal estocastico y con efecto de palanca subestiman los precios de las opciones exoticas. Un analisis heuristico sugiere que el diferente grado de fluctuacion de las trayectorias aleatorias de los modelos es el responsable de producir diferentes precios para las opciones de barrera.
En el segundo ensayo de esta tesis se examinan medidas de riesgo coherentes y funciones de distorsion concavas. Una familia de funciones concavas de distorsion es un conjunto de funciones concavas y crecientes, con dominio e imagen igual al intervalo unitario. Se usan las funciones de distorsion para definir medidas de riesgo coherentes. Demostramos que cualquier familia de funciones de distorsion que cumpla una ecuacion de traslacion, puede ser representada por una funcion de distribucion. Una aplicacion se puede encontrar en la ciencia actuarial: los principios de primas basados en los momentos son faciles de entender, pero en general no son monotonos y no se pueden utilizar para comparar los riesgos de diferentes contratos de seguros entre si. Nuestro teorema de representacion permite comparar dos riesgos de seguros entre si de acuerdo con un principio de primas basado en un momento, definiendo adecuadamente una medida de riesgo coherente.
En el ultimo ensayo de esta tesis, investigamos los mercados financieros con fricciones, donde los precios de compra y venta de instrumentos financieros se describen mediante funciones de precios sublineares. Estas funciones pueden definirse recursivamente utilizando medidas de riesgo coherentes. En un modelo binomial y en la presencia de costes de transaccion, demostramos la convergencia de los precios de compra y venta para varias opciones europeas y americanas, en particular opciones plain vanillas, asiaticas, lookback y de barrera. Realizamos varios experimentos numericos para confirmar los hallazgos teoricos. Aplicamos los resultados a los datos de mercado reales de las opciones plain vanilla europeas y americanas y calculamos una liquidez implicita para describir la diferencia de precios de compra y venta. Este metodo describe muy bien la liquidez en comparacion con el enfoque clasico de describir la diferencia entre los precios de compra y venta con las volatilidades implicitas de dichos precios.
##English Version##
This thesis consists of three essays. In the first essay, we test empirically the pricing performance of several advanced financial models. We calibrate six advanced stock price models to a time series of real market data of European options on the DAX, a German blue chip index. Via a Monte Carlo simulation, we price barrier down-and-out call options for all models and compare the modelled prices to given real market data of the barrier options. The Bates model reproduces barrier option prices well. The BNS model overvalues and Levy models with stochastic time-change and leverage undervalue the exotic options. A heuristic analysis suggests that the different degree of fluctuation of the random paths of the models are responsible of producing different prices for the barrier options.
The second essay of this thesis discusses the relationship between coherent risk measures and concave distortion functions. A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment based premium principle by defining an appropriate coherent risk measure.
In the last essay of this thesis, we investigate financial markets with frictions, where bid and ask prices of financial intruments are described by sublinear pricing functionals. Such functionals can be defined recursively using coherent risk measures. We prove the convergence of bid and ask prices for various European and American possible path-dependent options, in particular plain vanilla, Asian, lookback and barrier options in a binomial model in the presence of transaction costs. We perform several numerical experiments to confirm the theoretical findings. We apply the results to real market data of European and American plain vanilla options and compute an implied liquidity to describe the bid-ask spread. This method describes liquidity over time very well, compared to the classical approach of describing the bid-ask spread by quoting bid and ask implied volatilities.
6 citations
Cites background from "Lévy processes and infinitely divis..."
...A general introduction to Lévy processes can be found in Sato (1999), Schoutens (2003) and Applebaum (2009)....
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Posted Content•
TL;DR: In this high-dimensional semiparametric deconvolution problem, spectral thresholding estimators that are adaptive to the sparsity of $\Sigma$ are proposed and an oracle inequality for these estimators under model miss-specification is established.
Abstract: We study the estimation of the covariance matrix $\Sigma$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of $\Sigma$. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in $n/\log p$. We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.
6 citations
Additional excerpts
...10 in [45], which yields ∣∣ log |ψ0(u)|∣∣ 6 L ∑...
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TL;DR: The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view as discussed by the authors, where they considered a continuous-time stochastic volatility model with a general continuous volatility process and observed it under a high-frequency sampling scheme.
Abstract: This article gives an exhaustive mathematical analysis of the Gumbel test for additive jump components based on extreme value theory. The Gumbel test was first introduced by Lee and Mykland in 2008 from an economical point of view. They consider a continuous-time stochastic volatility model with a general continuous volatility process and observe it under a high-frequency sampling scheme. The test statistics based on the maximum of increments converges to the Gumbel distribution under the null hypothesis of no additive jump component and to infinity otherwise. Our article presents a moment method based technique that provides some deeper mathematical insights into the convergence and divergence case of the test statistics. In the non-jump case we are able to prove the convergence to the Gumbel distribution under greatly weak assumptions: The volatility process has to be merely pathwise Holder continuous with an arbitrary random Holder exponent and we have no restrictions concerning an additional d...
6 citations
Posted Content•
TL;DR: In this paper, a Hunt type formula for the infinitesimal generators of the Levy process on the quantum groups (SU_q(N) and U_qN) was proposed.
Abstract: We provide a Hunt type formula for the infinitesimal generators of Levy process on the quantum groups $SU_q(N)$ and $U_q(N)$. In particular, we obtain a decomposition of such generators into a gaussian part and a `jump type' part determined by a linear functional that resembles the functional induced by the Levy measure. The jump part on $SU_q(N)$ decomposes further into parts that live on the quantum subgroups $SU_q(n)$, $n\le N$. Like in the classical Hunt formula for locally compact Lie groups, the ingredients become unique once a certain projection is chosen. There are analogous result for $U_q(N)$.
6 citations
Cites methods from "Lévy processes and infinitely divis..."
...The domain may be thought of as those functions that possess a second order Taylor expansion around the neutral element of G, e. Hunt’s formula [15], a generalization of the Lévy-Khintchine formula for R (see for instance Applebaum [3] or Sato [21]), asserts that ψ(f) = ψG(f) + ∫ G\{e} [P(f)](g) dL(g)....
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...Hunt’s formula [15], a generalization of the Lévy-Khintchine formula for R (see for instance Applebaum [3] or Sato [21]), asserts that...
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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