Book Description
The Malliavin calculus (or stochastic calculus of variations) is an infinite-dimensional differential calculus on a Gaussian space. Originally, it was developed to provide a probabilistic proof to Hörmander's "sum of squares" theorem, but it has found a wide range of applications in stochastic analysis. This monograph presents the main features of the Malliavin calculus and discusses in detail its main applications. The author begins by developing the analysis on the Wiener space, and then uses this to establish the regularity of probability laws and to prove Hörmander's theorem. The regularity of the law of stochastic partial differential equations driven by a space-time white noise is also studied. The subsequent chapters develop the connection of the Malliavin with the anticipating stochastic calculus, studying anticipating stochastic differential equations and the Markov property of solutions to stochastic differential equations with boundary conditions. The second edition of this monograph includes recent applications of the Malliavin calculus in finance and a chapter devoted to the stochastic calculus with respect to the fractional Brownian motion.
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Diffusions and Elliptic Operators (Probability and its Applications)
Richard F. Bass
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Probabilistic Techniques in Analysis (Probability and its Applications)
ASIN: 0387983155 |
Book Description
This is author-approved bcc: This book discusses the interplay of diffusion processes and partial differential equations with an emphasis on probabilistic methods in PDE. It begins with stochastic differential equations, the probabilistic machinery needed to study PDE. After spending three chapters on probabilistic representations of solutions for PDE, regularity of solutions and one dimensional diffusions, the author discusses in depth two main types of second order linear differential operators: non-divergence operators and divergence operators, including topics such as the Harnack inequality of Krylov-Safonov for non-divergence operators and heat kernel estimates for divergence form operators. Martingale problems and the Malliavin calculus are presented in two other chapters. This book can be used as a textbook for a graduate course on diffusion theory with applications to PDE. It will also be a valuable reference to researchers in probability who are interested in PDE as well as for analysts who are interested in probabilistic methods. Richard F. Bass is Professor of Mathematics at the University of Washington. He has written many research papers on the topics covered by this book. Also Available: Richard F. Bass, Probabilistic Techniques in Analysis. Springer-Verlag New York, Inc, 0-387-94387-0
Book Description
Stochastic methods have become increasingly important in the analysis of a broad range of phenomena in natural sciences and economics. Many processes are described by differential equations where some of the parameters and/or the initial data are not known with complete certainty due to lack of information, uncertainty in the measurements, or incomplete knowledge of the mechanisms themselves. To compensate for this lack of information one introduces stochastic noise in the equations, either in the parameters or in the initial data which results in stochastic differential equations.
At the same time there has been considerable development in the mathematical theory of stochastic differential equations which are used to model these phenomena. In this book the authors give a comprehensive introduction to stochastic partial differential equations. Their approach is based on white noise analysis, where the often ill-defined white noise, the derivative of the familiar Brownian motion, is introduced rigorously as the fundamental object.
First some of the mathematical background is discussed to provide the necessary tools to study several different stochastic partial differential equations. The techniques are primarily derived from functional analysis. The Wiener-Itô chaos expansion as well as the Itô/Skorohod integrals are developed in this setting, and properties of the Wick product and the Hermite transform are proved. The first applications are given to stochastic ordinary differential equations, e.g., the Volterra equation.
The main emphasis of the book is on stochastic partial differential equations. First the stochastic Poisson equation and the stochastic transport equation are discussed. Next, the authors consider the stochastic Schrödinger equation as well as the stochastic heat equation. The nonlinear Burgers' equation with a stochastic source is discussed, and finally the stochastic pressure equation, as well as other important equations are treated. The white noise approach often allows for solutions given by explicit formulas in terms of expectations of certain auxiliary processes. The noise in the above examples are all of Gaussian white noise type, but in the end the authors also show how to adapt the analysis to SPDEs involving noise of Poissonian type.
Customer Reviews:
A new approach to stochastic partial differential equations.......2000-07-21
SUMMARY: This book presents a new approach to stochastic partial differential equations based on white noise analysis. The framework makes heavy use of functional analysis and its main starting point is the Wiener chaos expansion and analogous expansions on different functional spaces (Schwartz spaces).
A stochastic PDE is a PDE containing a random noise term, which may be additive or multiplicative. One of the problems when working with Stochastic PDEs is to define a notion of solution which is meaningfully extendable to the nonlinear case. Problems arises because the noise term is highly irregular: for each sample of the noise, one has a (nonlinear) PDE with a very irregular term in it. In physical terms, one may encounter "ultraviolet" divergences. So, one is first faced with an existence/ unicity problem for such equations. Additionally, one would like to describe probabilitic properties of such solutions.
The method proposed by the authors can be described as follows: first, one expands the noise term in the PDE using a Wiener chaos expansion. Truncating the expansion at a certain order n yields a "regularized" equation in which the noise is smoothened. This can be roughly described as an ultraviolet cutoff. The equation then has a unique solution in an appropriate functional space. The solution of the original SPDE is then defined as the sequence of truncated solutions. In some cases, this sequence may converge in some classical sense in an appropriate function space to a weak or strong solution defined in the usual sense. But, in general, this is not the case and the notion of solution defined by the authors may be different from classical notions.
Although the title contains the word 'modeling', it may look as the abstract definition of solution proposed by the authors may have little to do with the physical notion of solution. One feels a need for a justification why this definition of a solution is physically relevant at all, which I feel is lacking. The authors give some examples, such as the noisy Burgers equation and the Kardar-Parisi-Zhang equation, but the results predicted for the solutions seem to be different than the ones predicted for example by renormalization group analysis for example regarding the scaling exponents for KPZ. Also, it would be interesting to compare this notion of solution with more classical ones for example using the semigroup/ Green function approach.
The approach proposed bears a strong resemblance to ultraviolet regularization schemes used in renormalization group theory. In fact, this framework may be seenas a probabilistic setting for renormalization methods.Unfortunately there is little discussion of this point in the book.
The first chapters contain an interesting review of white noise expansions and chaos expansions, useful in their own interest.
Overall I recommend this book as interesting for researchers in mathematical and theoretical physics.
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Forward-Backward Stochastic Differential Equations and their Applications (Lecture Notes in Mathematics)
Jin Ma , and
Jiongmin Yong
Manufacturer: Springer
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ASIN: 3540659609 |
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This volume is a survey/monograph on the recently developed theory of forward-backward stochastic differential equations (FBSDEs). Basic techniques such as the method of optimal control, the "Four Step Scheme", and the method of continuation are presented in full. Related topics such as backward stochastic PDEs and many applications of FBSDEs are also discussed in detail. The volume is suitable for readers with basic knowledge of stochastic differential equations, and some exposure to the stochastic control theory and PDEs. It can be used for researchers and/or senior graduate students in the areas of probability, control theory, mathematical finance, and other related fields.
Book Description
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics. This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself based on a general Gaussian space, going from the simple, finite-dimensional setting to the infinite-dimensional one. The final three chapters discuss recent research on regularity of the solution of stochastic partial differential equations and the existence and smoothness of their probability laws. About the author: Marta Sanz-Solé is Professor at the Faculty of Mathematics, University of Barcelona. She is a leading member of the research group on stochastic analysis at Barcelona, and in 1998 she received the Narcis Monturiol Award of Scientific and Technological Excellence from the autonomous government of Catalonia.
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Meshfree Methods for Partial Differential Equations III (Lecture Notes in Computational Science and Engineering)
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ASIN: 3540462147 |
Book Description
Meshfree methods for the numerical solution of partial differential equations are becoming more and more mainstream in many areas of applications. Their flexiblity and wide applicability are attracting engineers, scientists, and mathematicians to this very dynamic research area. This volume represents the state of the art in meshfree methods. It consists of articles which address the different meshfree techniques, their mathematical properties and their application in applied mathematics, physics and engineering.
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Nonlinear Stochastic Evolution Problems in Applied Sciences (Mathematics and Its Applications)
N. Bellomo ,
Z. Brzezniak , and
L.M. de Socio
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ASIN: 0792320425 |
Book Description
This volume deals with the analysis of nonlinear evolution problems described by partial differential equations having random or stochastic parameters. The emphasis throughout is on the actual determination of solutions, rather than on proving the existence of solutions, although mathematical proofs are given when this is necessary from an applications point of view. The content is divided into six chapters.
Chapter 1 gives a general presentation of mathematical models in continuum mechanics and a description of the way in which problems are formulated. Chapter 2 deals with the problem of the evolution of an unconstrained system having random space-dependent initial conditions, but which is governed by a deterministic evolution equation. Chapter 3 deals with the initial-boundary value problem for equations with random initial and boundary conditions as well as with random parameters where the randomness is modelled by stochastic separable processes. Chapter 4 is devoted to the initial-boundary value problem for models with additional noise, which obey Ito-type partial differential equations. Chapter 5 is essential devoted to the qualitative and quantitative analysis of the chaotic behaviour of systems in continuum physics. Chapter 6 provides indications on the solution of ill-posed and inverse problems of stochastic type and suggests guidelines for future research. The volume concludes with an Appendix which gives a brief presentation of the theory of stochastic processes.
Examples, applications and case studies are given throughout the book and range from those involving simple stochasticity to stochastic illposed problems.
For applied mathematicians, engineers and physicists whose work involves solving stochastic problems.
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Nonlinear Stochastic PDE's: Hydrodynamic Limit and Burgers' Turbulence (The IMA Volumes in Mathematics and its Applications)
Manufacturer: Springer
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Binding: Hardcover
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Chaos & Systems
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ASIN: 0387946241 |
Book Description
This volume is a based on recent research which focuses on the area of nonlinear stochastic partial differential equations. The first section contains work on fundamental problems of hydrodynamic limit for particle systems and on random media. The second part groups together papers under the umbrella of the name "Burgers' turbulence", although a broader spectrum of stochastic problem for the Burgers' equation is actually addressed. Finally, the last part deals with the stochastic Navier-Stokes equation both from mathematical and physical perspective. This book is suitable for mathematicians and students.
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Probability and Partial Differential Equations in Modern Applied Mathematics (The IMA Volumes in Mathematics and its Applications)
Manufacturer: Springer
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ASIN: 0387258795 |
Book Description
The IMA Summer Program on Probability and Partial Differential Equations in Modern Applied Mathematics took place July 21-August 1, 2003. The program was devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and as a tool in modeling. There is a growing recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering.
A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in analysis, as well as offer new perspectives on the phenomena for modeling purposes. In addition, such approaches can be effective in sorting out multiple scale structure and in the development of both non-Monte Carlo as well as Monte Carlo type numerical methods.
There is also a growing recognition of a role in the inclusion of stochastic terms in the modeling of complex flows, and the addition of such terms has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations.
This volume consists of original contributions by researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. The volume will be useful to researchers and graduate students who are interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in engineering and sciences.
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Random Fields and Stochastic Partial Differential Equations (Mathematics and Its Applications)
Y.A. Rozanov
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ASIN: 0792349849 |
Book Description
This book considers some models described by means of partial differential equations and boundary conditions with chaotic stochastic disturbance. In a framework of stochastic partial differential equations an approach is suggested to generalise solutions of stochastic boundary problems. The main topic concerns probabilistic aspects with applications to the most well-known random fields models which are representative for the corresponding stochastic Sobolev spaces. This work assumes basic knowledge of general analysis and probability, such as Hilbert space methods, Schwartz distributions, and Fourier transforms.
Audience: This volume will be of interest to researchers and postgraduate students whose work involves probability theory, stochastic processes and partial differential equations.
Books:
- The Probabilistic Method (Wiley-Interscience Series in Discrete Mathematics and Optimization)
- The Statistical Analysis of Failure Time Data (Wiley Series in Probability and Statistics)
- Twistor Geometry and Field Theory (Cambridge Monographs on Mathematical Physics)
- Urban Stormwater Management Planning with Analytical Probabilistic Models
- A First Course in Modular Forms (Graduate Texts in Mathematics)
- A Guide to MATLAB: For Beginners and Experienced Users
- A Multigrid Tutorial
- A Primer of Ecology
- An Introduction to Complex Analysis in Several Variables (North-Holland Mathematical Library)
- An Introduction to Complex Analysis in Several Variables (North-Holland Mathematical Library)
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