Gaussian Hilbert Spaces

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Author: Svante Janson

Publisher: Cambridge University Press

ISBN: 9780521561280

Category: Mathematics

Page: 340

View: 466

This book treats the fundamental mathematical properties that hold for a family of Gaussian random variables.

Zeros of Gaussian Analytic Functions and Determinantal Point Processes

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Author: John Ben Hough,Manjunath Krishnapur ,Yuval Peres ,B\'alint Vir\'ag

Publisher: American Mathematical Soc.

ISBN: 0821843737

Category: Mathematics

Page: 154

View: 1301

The book examines in some depth two important classes of point processes, determinantal processes and ``Gaussian zeros'', i.e., zeros of random analytic functions with Gaussian coefficients. These processes share a property of ``point-repulsion'', where distinct points are less likely to fall close to each other than in processes, such as the Poisson process, that arise from independent sampling. Nevertheless, the treatment in the book emphasizes the use of independence: for random power series, the independence of coefficients is key; for determinantal processes, the number of points in a domain is a sum of independent indicators, and this yields a satisfying explanation of the central limit theorem (CLT) for this point count. Another unifying theme of the book is invariance of considered point processes under natural transformation groups. The book strives for balance between general theory and concrete examples. On the one hand, it presents a primer on modern techniques on the interface of probability and analysis. On the other hand, a wealth of determinantal processes of intrinsic interest are analyzed; these arise from random spanning trees and eigenvalues of random matrices, as well as from special power series with determinantal zeros. The material in the book formed the basis of a graduate course given at the IAS-Park City Summer School in 2007; the only background knowledge assumed can be acquired in first-year graduate courses in analysis and probability.

Stochastic Analysis and Applications to Finance

Essays in Honour of Jia-an Yan

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Author: Tusheng Zhang,Xunyu Zhou

Publisher: World Scientific

ISBN: 9814489158

Category: Mathematics

Page: 464

View: 9194

This volume is a collection of solicited and refereed articles from distinguished researchers across the field of stochastic analysis and its application to finance. The articles represent new directions and newest developments in this exciting and fast growing area. The covered topics range from Markov processes, backward stochastic differential equations, stochastic partial differential equations, stochastic control, potential theory, functional inequalities, optimal stopping, portfolio selection, to risk measure and risk theory. It will be a very useful book for young researchers who want to learn about the research directions in the area, as well as experienced researchers who want to know about the latest developments in the area of stochastic analysis and mathematical finance. Contents:Non-Linear Evolution Equations Driven by Rough Paths (Thomas Cass, Zhongmin Qian and Jan Tudor)Optimal Stopping Times with Different Information Levels and with Time Uncertainty (Arijit Chakrabarty and Xin Guo)Finite Horizon Optimal Investment and Consumption with CARA Utility and Proportional Transaction Costs (Yingshan Chen, Min Dai and Kun Zhao)MUniform Integrability of Exponential Martingales and Spectral Bounds of Non-Local Feynman-Kac Semigroups (Zhen-Qing Chen)Continuous-Time Mean-Variance Portfolio Selection with Finite Transactions (Xiangyu Cui, Jianjun Gao and Duan Li)Quantifying Model Uncertainties in the Space of Probability Measures (J Duan, T Gao and G He)A PDE Approach to Multivariate Risk Theory (Robert J Elliott, Tak Kuen Siu and Hailiang Yang)Stochastic Analysis on Loop Groups (Shizan Fang)Existence and Stability of Measure Solutions for BSDE with Generators of Quadratic Growth (Alexander Fromm, Peter Imkeller and Jianing Zhang)Convex Capital Requirements for Large Portfolios (Hans Föllmer and Thomas Knispel)The Mixed Equilibrium of Insider Trading in the Market with Rational Expected Price (Fuzhou Gong and Hong Liu)Some Results on Backward Stochastic Differential Equations Driven by Fractional Brownian Motions (Yaozhong Hu, Daniel Ocone and Jian Song)Potential Theory of Subordinate Brownian Motions Revisited (Panki Kim, Renming Song and Zoran Vondraček)Research on Social Causes of the Financial Crisis (Steven Kou)Wick Formulas and Inequalities for the Quaternion Gaussian and β-Permanental Variables (Wenbo V Li and Ang Wei)Further Study on Web Markov Skeleton Processes (Yuting Liu, Zhi-Ming Ma and Chuan Zhou)MLE of Parameters in the Drifted Brownian Motion and Its Error (Lemee Nakamura and Weian Zheng)Optimal Partial Information Control of SPDEs with Delay and Time-Advanced Backward SPDEs (Bernt Øksendal, Agnès Sulem and Tusheng Zhang)Simulation of Diversified Portfolios in Continuous Financial Markets (Eckhard Platen and Renata Rendek)Coupling and Applications (Feng-Yu Wang)SDEs and a Generalised Burgers Equation (Jiang-Lun Wu and Wei Yang)Mean-Variance Hedging in the Discontinuous Case (Jianming Xia) Readership: Graduates and researchers in stochatic analysis and mathematical finance. Keywords:Stochastic Analysis;Finance;Stochastic Partial Differential Equations;Backward Stochastic Differential Equations;Potential TheoryKey Features:Unique combination of stochastic analysis and financeSolicited articles from leading researchers in the areaA volume in honour of Jia-an Yan, a prominent scholar in both stochastic analysis and mathematical finance

Diffusion, Quantum Theory, and Radically Elementary Mathematics. (MN-47)

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Author: William G. Faris

Publisher: Princeton University Press

ISBN: 1400865255

Category: Mathematics

Page: 256

View: 9835

Diffusive motion--displacement due to the cumulative effect of irregular fluctuations--has been a fundamental concept in mathematics and physics since Einstein's work on Brownian motion. It is also relevant to understanding various aspects of quantum theory. This book explains diffusive motion and its relation to both nonrelativistic quantum theory and quantum field theory. It shows how diffusive motion concepts lead to a radical reexamination of the structure of mathematical analysis. The book's inspiration is Princeton University mathematics professor Edward Nelson's influential work in probability, functional analysis, nonstandard analysis, stochastic mechanics, and logic. The book can be used as a tutorial or reference, or read for pleasure by anyone interested in the role of mathematics in science. Because of the application of diffusive motion to quantum theory, it will interest physicists as well as mathematicians. The introductory chapter describes the interrelationships between the various themes, many of which were first brought to light by Edward Nelson. In his writing and conversation, Nelson has always emphasized and relished the human aspect of mathematical endeavor. In his intellectual world, there is no sharp boundary between the mathematical, the cultural, and the spiritual. It is fitting that the final chapter provides a mathematical perspective on musical theory, one that reveals an unexpected connection with some of the book's main themes.

Forthcoming Books

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Author: Rose Arny

Publisher: N.A

ISBN: N.A

Category: American literature

Page: N.A

View: 5616

Dynamics of Linear Operators

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Author: Frédéric Bayart,Étienne Matheron

Publisher: Cambridge University Press

ISBN: 0521514967

Category: Mathematics

Page: 337

View: 6501

The first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.

Ridge Functions

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Author: Allan Pinkus

Publisher: Cambridge University Press

ISBN: 1107124395

Category: Computers

Page: 218

View: 1057

Presents the state of the art in the theory of ridge functions, providing a solid theoretical foundation.

Parameter Estimation in Fractional Diffusion Models

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Author: Kęstutis Kubilius,Yuliya Mishura,Kostiantyn Ralchenko

Publisher: Springer

ISBN: 3319710303

Category: Mathematics

Page: 390

View: 8966

This book is devoted to parameter estimation in diffusion models involving fractional Brownian motion and related processes. For many years now, standard Brownian motion has been (and still remains) a popular model of randomness used to investigate processes in the natural sciences, financial markets, and the economy. The substantial limitation in the use of stochastic diffusion models with Brownian motion is due to the fact that the motion has independent increments, and, therefore, the random noise it generates is “white,” i.e., uncorrelated. However, many processes in the natural sciences, computer networks and financial markets have long-term or short-term dependences, i.e., the correlations of random noise in these processes are non-zero, and slowly or rapidly decrease with time. In particular, models of financial markets demonstrate various kinds of memory and usually this memory is modeled by fractional Brownian diffusion. Therefore, the book constructs diffusion models with memory and provides simple and suitable parameter estimation methods in these models, making it a valuable resource for all researchers in this field. The book is addressed to specialists and researchers in the theory and statistics of stochastic processes, practitioners who apply statistical methods of parameter estimation, graduate and post-graduate students who study mathematical modeling and statistics.

Concentration, Functional Inequalities, and Isoperimetry

International Workshop on Concentration, Functional Inequalities, and Isoperimetry, October 29-November 1, 2009, Florida Atlantic University, Boca Raton, Florida

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Author: Christian Houdré

Publisher: American Mathematical Soc.

ISBN: 0821849719

Category: Mathematics

Page: 211

View: 2245

The volume contains the proceedings of the international workshop on Concentration, Functional Inequalities and Isoperimetry, held at Florida Atlantic University in Boca Raton, Florida, from October 29-November 1, 2009. The interactions between concentration, isoperimetry and functional inequalities have led to many significant advances in functional analysis and probability theory. Important progress has also taken place in combinatorics, geometry, harmonic analysis and mathematical physics, to name but a few fields, with recent new applications in random matrices and information theory. This book should appeal to graduate students and researchers interested in the fascinating interplay between analysis, probability, and geometry.

Probability on Real Lie Algebras

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Author: Uwe Franz,Nicolas Privault

Publisher: Cambridge University Press

ISBN: 110712865X

Category: Mathematics

Page: 302

View: 5448

This monograph is a progressive introduction to non-commutativity in probability theory, summarizing and synthesizing recent results about classical and quantum stochastic processes on Lie algebras. In the early chapters, focus is placed on concrete examples of the links between algebraic relations and the moments of probability distributions. The subsequent chapters are more advanced and deal with Wigner densities for non-commutative couples of random variables, non-commutative stochastic processes with independent increments (quantum Lévy processes), and the quantum Malliavin calculus. This book will appeal to advanced undergraduate and graduate students interested in the relations between algebra, probability, and quantum theory. It also addresses a more advanced audience by covering other topics related to non-commutativity in stochastic calculus, Lévy processes, and the Malliavin calculus.

A Primer on the Dirichlet Space

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Author: Omar El-Fallah,Karim Kellay,Javad Mashreghi,Thomas Ransford

Publisher: Cambridge University Press

ISBN: 1107729777

Category: Mathematics

Page: 227

View: 8765

The Dirichlet space is one of the three fundamental Hilbert spaces of holomorphic functions on the unit disk. It boasts a rich and beautiful theory, yet at the same time remains a source of challenging open problems and a subject of active mathematical research. This book is the first systematic account of the Dirichlet space, assembling results previously only found in scattered research articles, and improving upon many of the proofs. Topics treated include: the Douglas and Carleson formulas for the Dirichlet integral, reproducing kernels, boundary behaviour and capacity, zero sets and uniqueness sets, multipliers, interpolation, Carleson measures, composition operators, local Dirichlet spaces, shift-invariant subspaces, and cyclicity. Special features include a self-contained treatment of capacity, including the strong-type inequality. The book will be valuable to researchers in function theory, and with over 100 exercises it is also suitable for self-study by graduate students.

Stochastic Partial Differential Equations for Computer Vision with Uncertain Data

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Author: Tobias Preusser,Robert M. Kirby,Torben Pätz

Publisher: Morgan & Claypool Publishers

ISBN: 1681731444

Category: Mathematics

Page: 160

View: 7362

In image processing and computer vision applications such as medical or scientific image data analysis, as well as in industrial scenarios, images are used as input measurement data. It is good scientific practice that proper measurements must be equipped with error and uncertainty estimates. For many applications, not only the measured values but also their errors and uncertainties, should be—and more and more frequently are—taken into account for further processing. This error and uncertainty propagation must be done for every processing step such that the final result comes with a reliable precision estimate. The goal of this book is to introduce the reader to the recent advances from the field of uncertainty quantification and error propagation for computer vision, image processing, and image analysis that are based on partial differential equations (PDEs). It presents a concept with which error propagation and sensitivity analysis can be formulated with a set of basic operations. The approach discussed in this book has the potential for application in all areas of quantitative computer vision, image processing, and image analysis. In particular, it might help medical imaging finally become a scientific discipline that is characterized by the classical paradigms of observation, measurement, and error awareness. This book is comprised of eight chapters. After an introduction to the goals of the book (Chapter 1), we present a brief review of PDEs and their numerical treatment (Chapter 2), PDE-based image processing (Chapter 3), and the numerics of stochastic PDEs (Chapter 4). We then proceed to define the concept of stochastic images (Chapter 5), describe how to accomplish image processing and computer vision with stochastic images (Chapter 6), and demonstrate the use of these principles for accomplishing sensitivity analysis (Chapter 7). Chapter 8 concludes the book and highlights new research topics for the future.

Random Graphs

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Author: Svante Janson,Tomasz Luczak,Andrzej Rucinski

Publisher: John Wiley & Sons

ISBN: 1118030966

Category: Mathematics

Page: 348

View: 3658

A unified, modern treatment of the theory of random graphs-including recent results and techniques Since its inception in the 1960s, the theory of random graphs has evolved into a dynamic branch of discrete mathematics. Yet despite the lively activity and important applications, the last comprehensive volume on the subject is Bollobas's well-known 1985 book. Poised to stimulate research for years to come, this new work covers developments of the last decade, providing a much-needed, modern overview of this fast-growing area of combinatorics. Written by three highly respected members of the discrete mathematics community, the book incorporates many disparate results from across the literature, including results obtained by the authors and some completely new results. Current tools and techniques are also thoroughly emphasized. Clear, easily accessible presentations make Random Graphs an ideal introduction for newcomers to the field and an excellent reference for scientists interested in discrete mathematics and theoretical computer science. Special features include: * A focus on the fundamental theory as well as basic models of random graphs * A detailed description of the phase transition phenomenon * Easy-to-apply exponential inequalities for large deviation bounds * An extensive study of the problem of containing small subgraphs * Results by Bollobas and others on the chromatic number of random graphs * The result by Robinson and Wormald on the existence of Hamilton cycles in random regular graphs * A gentle introduction to the zero-one laws * Ample exercises, figures, and bibliographic references