Helge Holden, Bernt Oksendal, Jan Uboe, Tusheng Zhang

Stochastic Partial Differential Equations (eBook, PDF)

A Modeling, White Noise Functional Approach

• Format: PDF

Produktbeschreibung

This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve th,~se SPDEs explicitly, or at least provide algorithms or approximations for the solutions.

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Produktdetails

• Verlag: Birkhäuser Boston
• Seitenzahl: 231
• Erscheinungstermin: 1. Dezember 2013
• Englisch
• ISBN-13: 9781468492156
• Artikelnr.: 44198966

Autorenporträt

Helge Holden is a professor of mathematics at the Norwegian University of Science and Technology and an adjunt professor at the Center of Mathematics for Applications, part of the University of Oslo.  He has done extensive research in stochastic analysis, in particular in its application to flow in porous media. Bernt Øksendal is a professor at the Center of Mathematics for Applications at the University of Oslo.  He is a winner of the Nansen Prize for research in stochastic analysis and its applications. Jan Ubøe is a professor in the Department of Finance and Management Sciences at the Norwegian School of Economics and Business Administration.  He has written many papers about this subject. Tusheng Zhang is a professor of probability at the University of Manchester.  His current area of research is stochastic differential and partial differential equations, and he recently published a monograph on fractional Brownian fields with Bernt Øksendal and others.

Inhaltsangabe

1. Introduction.
1.1. Modeling by stochastic differential equations.
2. Framework.
2.1. White noise.
2.2. The Wiener
Itô chaos expansion.
2.3. Stochastic test functions and stochastic distributions.
2.4. The Wick product.
2.5. Wick multiplication and Itô/Skorohod integration.
2.6. The Hermite transform.
2.7. The S)p,rN spaces and the S
transform.
2.8. The topology of (S)
1N.
2.9. The F
transform and the Wick product on L1 (?).
2.10. The Wick product and translation.
2.11. Positivity.
3. Applications to stochastic ordinary differential equations.
3.1. Linear equations.
3.2. A model for population growth in a crowded stochastic environment.
3.3. A general existence and uniqueness theorem.
3.4. The stochastic Volterra equation.
3.5. Wick products versus ordinary products: A comparison experiment Variance properties.
3.6. Solution and Wick approximation of quasilinear SDE.
4. Stochastic partial differential equations.
4.1. General remarks.
4.2. The stochastic Poisson equation.
4.3. The stochastic transport equation.
4.4. The stochastic Schrödinger equation.
4.5. The viscous Burgers' equation with a stochastic source.
4.6. The stochastic pressure equation.
4.7. The heat equation in a stochastic, anisotropic medium.
4.8. A class of quasilinear parabolic SPDEs.
4.9. SPDEs driven by Poissonian noise.
Appendix A. The Bochner
Minlos theorem.
Appendix B. A brief review of Itô calculus.
The Itô formula.
Stochastic differential equations.
The Girsanov theorem.
Appendix C. Properties of Hermite polynomials.
Appendix D. Independence of bases in Wick products.
References.
List of frequently used notation and symbols.

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Preface to the Second Edition.- Preface to the First Edition.- Introduction.- Framework.- Applications to stochastic ordinary differential equations.- Stochastic partial differential equations driven by Brownian white noise.- Stochastic partial differential equations driven by Lévy white noise.- Appendix A. The Bochner-Minlos theorem.- Appendix B. Stochastic calculus based on Brownian motion.- Appendix C. Properties of Hermite polynomials.- Appendix D. Independence of bases in Wick products.- Appendix E. Stochastic calculus based on Lévy processes- References.- List of frequently used notation and symbols.- Index.

Rezensionen

"The authors have made significant contributions to each of the areas. As a whole, the book is well organized and very carefully written and the details of the proofs are basically spelled out... This is a rich and demanding book... It will be of great value for students of probability theory or SPDEs with an interest in the subject, and also for professional probabilists." -Mathematical Reviews

"...a comprehensive introduction to stochastic partial differential equations." -Zentralblatt MATH

"This book will be invaluable to anyone interested in doing research in white noise theory or in applying this theory to solving real-world problems." -Computing Reviews