03657nam 22006015 450 991095717710332120250730101807.03-662-06400-610.1007/978-3-662-06400-9(CKB)2660000000027140(SSID)ssj0001296280(PQKBManifestationID)11766117(PQKBTitleCode)TC0001296280(PQKBWorkID)11348123(PQKB)11158428(DE-He213)978-3-662-06400-9(MiAaPQ)EBC3100326(PPN)188265317(EXLCZ)99266000000002714020130107d1999 u| 0engurnn#008mamaatxtccrContinuous Martingales and Brownian Motion /by Daniel Revuz, Marc Yor3rd ed. 1999.Berlin, Heidelberg :Springer Berlin Heidelberg :Imprint: Springer,1999.1 online resource (XIII, 602 p.)Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,2196-9701 ;293Bibliographic Level Mode of Issuance: Monograph3-540-64325-7 3-642-08400-1 Includes bibliographical references and indexes.0. Preliminaries -- I. Introduction -- II. Martingales -- III. Markov Processes -- IV. Stochastic Integration -- V. Representation of Martingales -- VI. Local Times -- VII. Generators and Time Reversal -- VIII. Girsanov’s Theorem and First Applications -- IX. Stochastic Differential Equations -- X. Additive Functionals of Brownian Motion -- XI. Bessel Processes and Ray-Knight Theorems -- XII. Excursions -- XIII. Limit Theorems in Distribution -- §1. Gronwall’s Lemma -- §2. Distributions -- §3. Convex Functions -- §4. Hausdorff Measures and Dimension -- §5. Ergodic Theory -- §6. Probabilities on Function Spaces -- §7. Bessel Functions -- §8. Sturm-Liouville Equation -- Index of Notation -- Index of Terms -- Catalogue.From the reviews: "This is a magnificent book! Its purpose is to describe in considerable detail a variety of techniques used by probabilists in the investigation of problems concerning Brownian motion. The great strength of Revuz and Yor is the enormous variety of calculations carried out both in the main text and also (by implication) in the exercises. ... This is THE book for a capable graduate student starting out on research in probability: the effect of working through it is as if the authors are sitting beside one, enthusiastically explaining the theory, presenting further developments as exercises, and throwing out challenging remarks about areas awaiting further research..." Bull.L.M.S. 24, 4 (1992) Since the first edition in 1991, an impressive variety of advances has been made in relation to the material of this book, and these are reflected in the successive editions.Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics,2196-9701 ;293ProbabilitiesProbability TheoryProbabilities.Probability Theory.519.2/8760G44msc60J60mscRevuz Danielauthttp://id.loc.gov/vocabulary/relators/aut54759Yor Marcauthttp://id.loc.gov/vocabulary/relators/autMiAaPQMiAaPQMiAaPQBOOK9910957177103321Continuous martingales and Brownian motion375923UNINA05892nam 22005295 450 991025431060332120200704092820.03-319-57511-210.1007/978-3-319-57511-7(DE-He213)978-3-319-57511-7(MiAaPQ)EBC5017527(PPN)204534151(CKB)4100000000586877(EXLCZ)99410000000058687720170901d2017 u| 0engurnn|008mamaatxtrdacontentcrdamediacrrdacarrierNumerical Methods for Stochastic Partial Differential Equations with White Noise /by Zhongqiang Zhang, George Em Karniadakis1st ed. 2017.Cham :Springer International Publishing :Imprint: Springer,2017.1 online resource (XV, 394 p. 36 illus., 34 illus. in color.) Applied Mathematical Sciences,0066-5452 ;196Preface -- Prologue -- Brownian Motion and Stochastic Calculus -- Numerical Methods for Stochastic Differential Equations -- Part I Stochastic Ordinary Differential Equations -- Numerical Schemes for SDEs with Time Delay Using the Wong-Zakai Approximation -- Balanced Numerical Schemes for SDEs with non-Lipschitz Coefficients -- Part II Temporal White Noise -- Wiener Chaos Methods for Linear Stochastic Advection-Diffusion-Reaction Equations -- Stochastic Collocation Methods for Differential Equations with White Noise -- Comparison Between Wiener Chaos Methods and Stochastic Collocation Methods -- Application of Collocation Method to Stochastic Conservation Laws -- Part III Spatial White Noise -- Semilinear Elliptic Equations with Additive Noise -- Multiplicative White Noise: The Wick-Malliavin Approximation -- Epilogue -- Appendices -- A. Basics of Probability -- B. Semi-analytical Methods for SPDEs -- C. Gauss Quadrature -- D. Some Useful Inequalities and Lemmas -- E. Computation of Convergence Rate.This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical comparison with other integration methods in random space is made. Part III covers spatial white noise. Here the authors discuss numerical methods for nonlinear elliptic equations as well as other equations with additive noise. Numerical methods for SPDEs with multiplicative noise are also discussed using the Wiener chaos expansion method. In addition, some SPDEs driven by non-Gaussian white noise are discussed and some model reduction methods (based on Wick-Malliavin calculus) are presented for generalized polynomial chaos expansion methods. Powerful techniques are provided for solving stochastic partial differential equations. This book can be considered as self-contained. Necessary background knowledge is presented in the appendices. Basic knowledge of probability theory and stochastic calculus is presented in Appendix A. In Appendix B some semi-analytical methods for SPDEs are presented. In Appendix C an introduction to Gauss quadrature is provided. In Appendix D, all the conclusions which are needed for proofs are presented, and in Appendix E a method to compute the convergence rate empirically is included. In addition, the authors provide a thorough review of the topics, both theoretical and computational exercises in the book with practical discussion of the effectiveness of the methods. Supporting Matlab files are made available to help illustrate some of the concepts further. Bibliographic notes are included at the end of each chapter. This book serves as a reference for graduate students and researchers in the mathematical sciences who would like to understand state-of-the-art numerical methods for stochastic partial differential equations with white noise.Applied Mathematical Sciences,0066-5452 ;196Numerical analysisProbabilitiesDifferential equations, PartialNumerical Analysishttps://scigraph.springernature.com/ontologies/product-market-codes/M14050Probability Theory and Stochastic Processeshttps://scigraph.springernature.com/ontologies/product-market-codes/M27004Partial Differential Equationshttps://scigraph.springernature.com/ontologies/product-market-codes/M12155Numerical analysis.Probabilities.Differential equations, Partial.Numerical Analysis.Probability Theory and Stochastic Processes.Partial Differential Equations.518Zhang Zhongqiangauthttp://id.loc.gov/vocabulary/relators/aut766754Karniadakis George Emauthttp://id.loc.gov/vocabulary/relators/autBOOK9910254310603321Numerical Methods for Stochastic Partial Differential Equations with White Noise2275306UNINA