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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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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....
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...The interested reader is referred to Lukacs (1970) or Sato (1999), to name but two of many possible references....
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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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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
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
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TL;DR: In this article, the exact asymptotics for the distribution of the first time a L\'evy process $X_t$ crosses a negative level $-x was studied.
Abstract: We study the exact asymptotics for the distribution of the first time $\tau_x$ a L\'evy process $X_t$ crosses a negative level $-x$. We prove that $\mathbf P(\tau_x>t)\sim V(x)\mathbf P(X_t\ge 0)/t$ as $t\to\infty$ for a certain function $V(x)$. Using known results for the large deviations of random walks we obtain asymptotics for $\mathbf P(\tau_x>t)$ explicitly in both light and heavy tailed cases. We also apply our results to find asymptotics for the distribution of the busy period in an M/G/1 queue.
6 citations
TL;DR: In this paper, the authors explore the connection between increasing and logconcave reward functions and optimal stopping rules of threshold form, and show that if a reward function defined on Z is nonnegative, increasing, logcon-cave, then the optimal stopping rule is of threshold-form provided the underlying random walk is skip-free to the right.
Abstract: In the literature, the problem of maximizing the expected discounted reward over all stopping rules has been explicitly solved for a number of reward functions (including (maxfx;0g) , > 0, in particular) when the underlying process is either a random walk in discrete time or a Levy process in continuous time. All of such reward functions are increasing and logconcave while the corresponding optimal stopping rules have the threshold form. In this paper, we explore the close connection between increasing and logconcave reward functions and optimal stopping rules of threshold form. In the discrete case, we show that if a reward function defined on Z is nonnegative, increasing and logconcave, then the optimal stopping rule is of threshold form provided the underlying random walk is skip-free to the right. In the continuous case, it is shown that for a reward function defined on R which is nonnegative, increasing, logconcave and right-continuous, the optimal stopping rule is of threshold form provided the underlying process is a spectrally negative Levy process. Furthermore, we also establish the necessity of logconcavity and monotonicity of a reward function in order for the optimal stopping rule to be of threshold form in the discrete (continuous, resp.) case when the underlying process belongs to the class of Bernoulli random walks (Brownian motions, resp.) with a downward drift. These results together provide a partial characterization of the threshold structure of optimal stopping rules.
6 citations
DOI•
01 Jan 2016
TL;DR: A proper statistically-based discretization paradigm for inverse problems is introduced and a practical phase reconstruction algorithm that incorporates a sparsity-based regularization is developed.
Abstract: We propose new regularization models to solve inverse problems encountered in biomedical imaging applications. In formulating mathematical schemes, we base our approach on the sparse signal processing principles that have emerged as a central paradigm in the field. We adopt a variational perspective and specify the proposed sparsity-promoting data reconstruction models as energy minimization problems. To design practical algorithms, we develop novel iterative methods to efficiently perform the consequent optimization task. The thesis is organized in three main parts. In the first part, our main contribution is the introduction of a proper statistically-based discretization paradigm for inverse problems. In particular, our framework considers a continuous-domain stochastic signal model and characterizes a specific class of inference algorithms. We show that derived inference-based methods cover the classical Tikhonov-type techniques as well as a wide range of the sparsity-promoting schemes including the well-known l1-norm regularization and its nonconvex variants. This provides a unifying stochastic perception of the resolution of inverse problems. In the second part, we propose a novel phase retrieval algorithm for imaging unstained biological samples that are optically thin. In specific terms, we use the transport-of-intensity equation (TIE) relating the spatial phase map of a field to the derivative of its intensity along the propagation direction. We analyze the implications of using the TIE formalism with finer and coarser approximations of said derivative. Based on this analysis, our contribution is a practical phase reconstruction algorithm that incorporates a sparsity-based regularization. The developed technique operates with a standard bright-field microscope. Experiments on real data illustrate that our phase reconstruction algorithm is viable and can be a low-cost alternative to dedicated phase microscopes. In the last part, we develop regularization schemes for vector fields, which have an increasing prevalence in medical imaging. In this context, our first contribution is a new regularizer that imposes sparsity on the singular values of the Jacobian of a given vector field. We show that the proposed regularization functional is a valid extension of total variation (TV) regularization to vector-valued functions. We utilize the developed framework for enhancing the streamline visualizations of experimentally acquired 4D flow MRI data. Since vector field regularization requires processing large volumes of multidimensional data, our second contribution is the development of a non-iterative denoising algorithm. In particular, we design model-based tight wavelet frames that are able to remove the spurious divergence content from the vector field, which is of interest in aortic blood flow imaging.
6 citations
TL;DR: In this article, it was shown that the law of a polynomial process is not determined by its generator, but by a combination of smoothness of the symbol and ellipticity.
Abstract: The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process is not necessarily determined by its generator. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.
6 citations
DOI•
01 Jan 2015
TL;DR: This thesis adopts a statistical perspective and model the signal as a realization of a stochastic process that exhibits sparsity as its central property, and proposes a novel nonlinear forward model and a corresponding algorithm for the quantitative estimation of the refractive index distribution of an object.
Abstract: This thesis addresses statistical inference for the resolution of inverse problems. Our work is motivated by the recent trend whereby classical linear methods are being replaced by nonlinear alternatives that rely on the sparsity of naturally occurring signals. We adopt a statistical perspective and model the signal as a realization of a stochastic process that exhibits sparsity as its central property. Our general strategy for solving inverse problems then lies in the development of novel iterative solutions for performing the statistical estimation. The thesis is organized in five main parts. In the first part, we provide a general overview of statistical inference in the context of inverse problems. We discuss wavelet--based and gradient--based algorithms for linear and nonlinear forward models. In the second part, we present an in-depth discussion of cycle spinning, which is a technique used to improve the quality of signals recovered with wavelet--based methods. Our main contribution here is its proof of convergence; we also introduce a novel consistent cycle-spinning algorithm for denoising statistical signals. In the third part, we introduce a stochastic signal model based on L\'evy processes and investigate popular gradient--based algorithms such as those that deploy total-variation regularization. We develop a novel algorithm based on belief propagation for computing the minimum mean-square error estimator and use it to benchmark several popular methods that recover signals with sparse derivatives. In the fourth part, we propose and analyze a novel adaptive generalized approximate message passing (adaptive GAMP) algorithm that reconstructs signals with independent wavelet-coefficients from generalized linear measurements. Our algorithm is an extension of the standard GAMP algorithm and allows for the joint learning of unknown statistical parameters. We prove that, when the measurement matrix is independent and identically distributed Gaussian, our algorithm is asymptotically consistent. This means that it performs as well as the oracle algorithm, which knows the parameters exactly. In the fifth and final part, we apply our methodology to an inverse problem in optical tomographic microscopy. In particular, we propose a novel nonlinear forward model and a corresponding algorithm for the quantitative estimation of the refractive index distribution of an object.
6 citations