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Stochastic Analysis and Random Maps in Hilbert Space

Stochastic Analysis and Random Maps in Hilbert Space
Author: A. A. Dorogovtsev
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 116
Release: 2019-01-14
Genre: Mathematics
ISBN: 3110618141

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This book is devoted to stochastic operators in Hilbert space. A number of models in modern probability theory apply the notion of a stochastic operator in explicit or latent form. In this book, objects from the Gaussian case are considered. Therefore, it is useful to consider all random variables and elements as functionals from the Wiener process or its formal derivative, i.e. white noise. The book consists of five chapters. The first chapter is devoted to stochastic calculus and its main goal is to prepare the tools for solving stochastic equations. In the second chapter the structure of stochastic equations, mainly the structure of Gaussian strong linear operators, is studied. In chapter 3 the definition of the action of the stochastic operator on random elements in considered. Chapter 4 deals with the mathematical models in which the notions of stochastic calculus arise and in the final chapter the equation with random operators is considered.


Stochastic Analysis and Mathematical Physics

Stochastic Analysis and Mathematical Physics
Author: A.B. Cruzeiro
Publisher: Springer Science & Business Media
Total Pages: 162
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461201276

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This volume represents the outgrowth of an ongoing workshop on stochastic analysis held in Lisbon. The nine survey articles in the volume extend concepts from classical probability and stochastic processes to a number of areas of mathematical physics. It is a good reference text for researchers and advanced students in the fields of probability, stochastic processes, analysis, geometry, mathematical physics, and physics. Key topics covered include: nonlinear stochastic wave equations, completely positive maps, Mehler-type semigroups on Hilbert spaces, entropic projections, and many others.


Hilbert And Banach Space-valued Stochastic Processes

Hilbert And Banach Space-valued Stochastic Processes
Author: Yuichiro Kakihara
Publisher: World Scientific
Total Pages: 539
Release: 2021-07-29
Genre: Mathematics
ISBN: 9811211760

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This is a development of the book entitled Multidimensional Second Order Stochastic Processes. It provides a research expository treatment of infinite-dimensional stationary and nonstationary stochastic processes or time series, based on Hilbert and Banach space-valued second order random variables. Stochastic measures and scalar or operator bimeasures are fully discussed to develop integral representations of various classes of nonstationary processes such as harmonizable, V-bounded, Cramér and Karhunen classes as well as the stationary class. A new type of the Radon-Nikodým derivative of a Banach space-valued measure is introduced, together with Schauder basic measures, to study uniformly bounded linearly stationary processes.Emphasis is on the use of functional analysis and harmonic analysis as well as probability theory. Applications are made from the probabilistic and statistical points of view to prediction problems, Kalman filter, sampling theorems and strong laws of large numbers. Generalizations are made to consider Banach space-valued stochastic processes to include processes of pth order for p ≥ 1. Readers may find that the covariance kernel is always emphasized and reveals another aspect of stochastic processes.This book is intended not only for probabilists and statisticians, but also for functional analysts and communication engineers.


Stochastic Analysis for Gaussian Random Processes and Fields

Stochastic Analysis for Gaussian Random Processes and Fields
Author: Vidyadhar S. Mandrekar
Publisher: CRC Press
Total Pages: 200
Release: 2015-06-23
Genre: Mathematics
ISBN: 1498707823

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Stochastic Analysis for Gaussian Random Processes and Fields: With Applications presents Hilbert space methods to study deep analytic properties connecting probabilistic notions. In particular, it studies Gaussian random fields using reproducing kernel Hilbert spaces (RKHSs).The book begins with preliminary results on covariance and associated RKHS


Stochastic Analysis

Stochastic Analysis
Author: Paul Malliavin
Publisher: Springer
Total Pages: 346
Release: 2015-06-12
Genre: Mathematics
ISBN: 3642150748

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In 5 independent sections, this book accounts recent main developments of stochastic analysis: Gross-Stroock Sobolev space over a Gaussian probability space; quasi-sure analysis; anticipate stochastic integrals as divergence operators; principle of transfer from ordinary differential equations to stochastic differential equations; Malliavin calculus and elliptic estimates; stochastic Analysis in infinite dimension.


Hilbert Space Methods in Probability and Statistical Inference

Hilbert Space Methods in Probability and Statistical Inference
Author: Christopher G. Small
Publisher: John Wiley & Sons
Total Pages: 268
Release: 2011-09-15
Genre: Mathematics
ISBN: 1118165535

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Explains how Hilbert space techniques cross the boundaries into the foundations of probability and statistics. Focuses on the theory of martingales stochastic integration, interpolation and density estimation. Includes a copious amount of problems and examples.


Infinite Dimensional Stochastic Analysis

Infinite Dimensional Stochastic Analysis
Author: Hui-Hsiung Kuo
Publisher: World Scientific
Total Pages: 257
Release: 2008
Genre: Science
ISBN: 981277954X

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This volume contains current work at the frontiers of research in infinite dimensional stochastic analysis. It presents a carefully chosen collection of articles by experts to highlight the latest developments in white noise theory, infinite dimensional transforms, quantum probability, stochastic partial differential equations, and applications to mathematical finance. Included in this volume are expository papers which will help increase communication between researchers working in these areas. The tools and techniques presented here will be of great value to research mathematicians, graduate students and applied mathematicians.


Stochastics in Finite and Infinite Dimensions

Stochastics in Finite and Infinite Dimensions
Author: Takeyuki Hida
Publisher: Springer Science & Business Media
Total Pages: 436
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461201675

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During the last fifty years, Gopinath Kallianpur has made extensive and significant contributions to diverse areas of probability and statistics, including stochastic finance, Fisher consistent estimation, non-linear prediction and filtering problems, zero-one laws for Gaussian processes and reproducing kernel Hilbert space theory, and stochastic differential equations in infinite dimensions. To honor Kallianpur's pioneering work and scholarly achievements, a number of leading experts have written research articles highlighting progress and new directions of research in these and related areas. This commemorative volume, dedicated to Kallianpur on the occasion of his seventy-fifth birthday, will pay tribute to his multi-faceted achievements and to the deep insight and inspiration he has so graciously offered his students and colleagues throughout his career. Contributors to the volume: S. Aida, N. Asai, K. B. Athreya, R. N. Bhattacharya, A. Budhiraja, P. S. Chakraborty, P. Del Moral, R. Elliott, L. Gawarecki, D. Goswami, Y. Hu, J. Jacod, G. W. Johnson, L. Johnson, T. Koski, N. V. Krylov, I. Kubo, H.-H. Kuo, T. G. Kurtz, H. J. Kushner, V. Mandrekar, B. Margolius, R. Mikulevicius, I. Mitoma, H. Nagai, Y. Ogura, K. R. Parthasarathy, V. Perez-Abreu, E. Platen, B. V. Rao, B. Rozovskii, I. Shigekawa, K. B. Sinha, P. Sundar, M. Tomisaki, M. Tsuchiya, C. Tudor, W. A. Woycynski, J. Xiong.


Gaussian Hilbert Spaces

Gaussian Hilbert Spaces
Author: Svante Janson
Publisher: Cambridge University Press
Total Pages: 358
Release: 1997-06-12
Genre: Mathematics
ISBN: 0521561280

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This book treats the very special and fundamental mathematical properties that hold for a family of Gaussian (or normal) random variables. Such random variables have many applications in probability theory, other parts of mathematics, statistics and theoretical physics. The emphasis throughout this book is on the mathematical structures common to all these applications. This will be an excellent resource for all researchers whose work involves random variables.


Stochastic Processes and Functional Analysis

Stochastic Processes and Functional Analysis
Author: Alan C. Krinik
Publisher: CRC Press
Total Pages: 526
Release: 2004-03-23
Genre: Mathematics
ISBN: 9780203913574

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This extraordinary compilation is an expansion of the recent American Mathematical Society Special Session celebrating M. M. Rao's distinguished career and includes most of the presented papers as well as ancillary contributions from session invitees. This book shows the effectiveness of abstract analysis for solving fundamental problems of stochas