Book Description
This book is designed as a text for graduate courses in stochastic processes. It is written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and these in turn permit a presentation of recent advances in financial economics (option pricing and consumption/investment optimization).
This book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The text is complemented by a large number of problems and exercises.
Customer Reviews:
A Superb Book.......2006-03-04
I found this book to be an excellent introduction into the subject matter. A good background in measure theoretic probability theory definitely helps, but even without much background, it is possible to understand all, but the finest measure theoretic points (I am a hobby mathematician with an engineering background, and I simply used the book "Probability Theory" by Laha & Rohatgi to learn what was needed about measure theory).
It is amazing, how the authors motivate, what they are doing using very few, but the right, words.
The pace of the book is just right, not too brisk and not too leasurely.
The only negative point is the following:
It takes some getting used to, that many important results are presented in the form of "problems". The solutions are generally given at the end of each chapter, so one has to thumb back an forth through the text.
Last but not least, the book contains virtually no misprints! For someone, who uses this book for self study, this is a very important point!
Massive Exercise to the Reader.......2005-03-31
This book isn't really the place to start learning about stochastic calculus. Get Oskendal's "Stochastic Differential Equations' for this.
Even to the prepared reader, this book is exasperating. It is as if the authors came up with an excellent outline for an advanced treatment of this topic. Then they realized that to do all of the material justice, they'd need to have not one, but two 400 page volumes. Their publisher must have balked at that idea, so their solution was to leave out half the detail, forcing each of our poor readers to re-generate the missing 400 pages of needed detail on his/her own. In the opinion of this reviewer, that is exactly what they have done with this text.
Fortunately for us all, there exists a nice two volume (800 page total pages) treatment of this material. Rogers & Williams "Diffusions, Markov Processes and Martingales" provides a thorough, accessible exposition with all the needed rigor, generality and detail.
Karatzas & Shreve's treatment of early foundational material is less than helpful to the student. Consider a pair of key results on martingales early on in the text: the optional sampling theorem and the optional stopping theorem. The authors "prove" the optional sampling theorem by appealing to the discrete time results in Chung's "A Course in Probability Theory" text and then applying limiting arguments to bootstrap to the continuous time case. Since all of the real "ideas" are in the discrete time case, it's not clear how much of a service the authors' treatment really is. Worse yet, the optional stopping theorem isn't even called out as a theorem, but instead buried as problem.
It is curious to see which topics inspire the authors to spill ink. For example in Chapter 2, we get not one, but 3, yes three different constructions of Brownian motion: convolved heat kernels, Haar interpolation and random walks/Wiener measure. Of course, only the last construction is used going forward and the first two constructions are not brimming over with detail. This is a curious indulgence in a text that is purposefully being stingy with detail. Our poor reader has to pay the price for this indulgence with an extremely terse treatment of the strong Markov property and reflection principle, the Blumenthal Zero-One Law, and other foundational properties of Brownian motion.
Chapter 3 represents the core of the text and develops all the of "greatest hits" including the Ito Integral, Ito's rule, Levy's characterization of Brownian motion, the martingale representation theorem, the Girsanov Theorem and an introduction of Brownian local time. (Brownian local time is further developed in Chapter 6). The development of the Ito Integral is shamelessly sketchy. All the theorems are correctly stated, but the "proofs" offered aren't detailed enough to explain why all of the stated assumptions are needed. When the reader gets to the development of Ito's rule, he/she finds a rude 3 sentence introduction to semi-martingales, a topic which hadn't been explored and never gets more than a passing mention in the authors' text.
Assuming that you've understood everything going on in the text up to this point, Chapter 4 is quite nice. It gives a very intuitive introduction in the role of the Mean Value Theorem as a hook connecting stochastic integrals with classical PDE's. The section on Harmonic functions and the Dirichlet problem is quite nice. The material on the heat equation requires properties of Brownian motion most easily derived from the convolved heat kernels construction. The chapter winds up with a nice treatment of the Feynman-Kac formulas.
After the PDE's material, the reader might develop a sense of hope that the remainder of the exposition will be readily accessible. This is not the case and with the SDE's in Chapter 5, the authors return to their now too familiar terse style as they study strong and weak solutions to stochastic differential equations. At one point, the authors decide to approach the problem by generalizing from functions to functionals without even so much as defining their notion of a functional.
Really, the only good role for this text is as base material for a do-it-yourself "Moore Method" class on stochastic calculus, like they used to do for general topology at the University of Texas. If you completed a Moore-style class this way and wrote up all of your work, you'd have a very fine text covering diffusions, Markov processes, and martingales
A Must.......2005-01-15
If you want to learn about stochastic calculus, this is the gold standard. Certainly a challenge, but if you can answer all the questions posed in the book you will have a very thorough knowledge of BM, stochastic integration with respect to BM, SDE's, and the SDE/PDE relationship.
While this is a great book, I do have a couple complaints. First of all there a points in the book where overly complicated notation can obscure the point being made. Secondly, since the book does not cover semimartingale integration, topics like the quadratic variation process (which are still important for things like representing a martingale as a time-changed BM) are spread throughout the book and can be difficult to find.
I recommend reading Rogers&Williams "Diffusions, Makov Processes, and Martingales" Vols I&II in addition to this book. Not only does it give a second (and sometimes easier) point of view on the somewhat difficult topic of stochastic analysis, but also covers some topics in greater generality including integration wrt a general semimartingale.
Finally, a warning to those who have interest but are not proper mathematics students. This book presupposes a fair amount of mathematical maturity - if you don't have a good understanding of real analysis and measure-theoretic probability you probably won't understand anything.
The best introduction.......2001-06-18
The theory of Brownian motion is ubiquitous in physics and mathematics, and has recently become very important in mathematical finance and network modeling. The observation of the irregular movement of pollen suspended in water by Robert Brown in 1828 led Albert Einstein to formulate a theory for Brownian motion. In this book the authors outline rigorously the theory of Browian motion. Their logic is impeccable, and the content is fascinating reading, even to those very experienced in the subject.
The authors begin in chapter 1 with the task of defining martingales and filtrations, with the notion of a stochastic process being adapted to a filtration taking on particular importance. They omit the proof that a process is progressively measurable if and only if it is measurable and adapted, because of the difficulty of the proof, but give a reference where the proof can be found. Continuous-time martingales are defined, with (compensated) Poisson processes given as an example. The Doob-Meyer decomposition and square-integrable martingales are discussed, and the chapter if full of exercises, with solutions provided to some of these at the end of the chapter. Brownian motion is formally defined in the next chapter, with its existence proven using Wiener measure on the space of continuous functions on the positive half line. The discussion in this chapter has to rank as one of the best in print, due to the meticulous and precise manner in which the material is presented. The Markov property of Brownian motion is proven, along with a good presentation of the Levi modulus of continuity. Readers working in constructive quantum field theory will see their usual construction of Wiener measure in the second exercise of the chapter. Those working in that area are used to seeing (conditional) Wiener measure defined on a collection of cylinder sets, which is then extended to the Borel subsets . Such a construction is done in this book, but the approach is somewhat different than what physicists normally see in quantum field theory.
The theory of stochastic integration is presented in Chapter 3, and it is superbly written. The authors are careful to distinguish the theory of integration for stochastic processes from the ordinary one with emphasis on the actual computation of stochastic integrals. The reader is first asked to explore the Stratonovitch and Ito integrals in an exercise., and then a thorough treatment is given by the authors later in the chapter. The authors point out the differences between the Ito and Stratonovich integrals, with the latter being defined for a smaller class of functions than the former. The important Ito rule for changing variables is discussed, and then used to give the Kunita-Watanabe martingale characterization of Brownian motion. Physicists involved in constructive quantum field theory will appreciate the discussion of the Trotter existence theorem in this chapter.
The connection of Brownian motion with partial differential equations, so familiar to physicists via the heat equation, is the subject of the next chapter. These equations give the transition probabilities of the stochastic process, and are studied here first in the context of harmonic analysis, namely the classical Dirichlet problem. This is followed by a beautiful treatment of the one-dimensional heat equation and the Feynman-Kac formulas. Those readers working in constructive quantum field theory will see the Green's function lurking in the background.
The very important topic of stochastic differential equations is outlined in chapter 5, with emphasis placed on the study of diffusive processes. The solutions of these equations have an immense literature, and the authors do not of course overview all of it, but do give a useful introduction. Both strong and weak solutions are discussed, with the Girsanov and Yamada-Watanabe techniques used throughout. Explicit solutions are given for linear stochastic differential equations, such as the Ornstein-Uhlenbeck process governing the Brownian motion of a particle with friction. Financial engineers will appreciate the discussion of the applications of this formalism to option pricing and the Merton consumption theory in this chapter. Options pricing is cast in martingale terms, and then the usual Black-Scholes equation is derived from this. The notorious Hamilton-Jacobi-Bellman equation is discussed in the consumption/investment problem, and the authors show how to employ techniques for solving this problem instead of solving this difficult nonlinear equation. The authors give a hint of the important Malliavin calculus in the Appendix and give references for the reader.
The last chapter of the book is more specialized than the rest and deals with the Levy theory of Brownian local time. This theory does have a connection with the theory of jump processes, which are currently very important in financial and network modeling. The authors do a fine job of explaining how Poisson random measures permit the event bookkeeping in these jump processes. Their discussion is applied to the computing of the transition probabilities for a Brownian motion with two-valued drift.
A rigorous but difficult presentation of SDEs.......1999-11-09
I recommend this book to anyone who wants to develop a deep understanding of Browninan Motion and Stochastic Calculus. However, the level of detail and rigor can obscure the main ideas, and so it is a very difficult introductory text for readers without a strong background in probability theory and continuous Markov processes. As a teaching assistant in a Mathematical Finance Masters program, I recommend that my students read Oskendal's Stochastic Differential Equations first, which gives an excellent introduction to the material without sacrificing rigor.
Average customer rating:
|
Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion
M. A. Lifshits , and
Werner Linde
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Mass Market Paperback
 General
| Education
| Nonfiction
| Subjects
| Books
General
| Science
| Subjects
| Books
Calculus
| Pure Mathematics
| Mathematics
| Science
| Subjects
| Books
General
| Mathematics
| Science
| Subjects
| Books
Probability & Statistics
| Applied
| Mathematics
| Science
| Subjects
| Books
Entropy
| Physics
| Science
| Subjects
| Books
Calculus
| Pure Mathematics
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
Stochastic Modeling
| Applied
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
General
| Education
| Professional & Technical
| Subjects
| Books
ASIN: 082182791X |
Book Description
Stochastic calculus and excursion theory are very efficient tools to obtain either exact or asymptotic results about Brownian motion and related processes. The emphasis of this book is on special classes of such Brownian functionals as:
- Gaussian subspaces of the Gaussian space of Brownian motion;
- Brownian quadratic funtionals;
- Brownian local times,
- Exponential functionals of Brownian motion with drift;
- Winding number of one or several Brownian motions around one or several points or a straight line, or curves;
- Time spent by Brownian motion below a multiple of its one-sided supremum.
Besides its obvious audience of students and lecturers the book also addresses the interests of researchers from core probability theory out to applied fields such as polymer physics and mathematical finance.
Book Description
Introductory Probability is a pleasure to read and provides a fine answer to the question: How do you construct Brownian motion from scratch, given that you are a competent analyst?
There are at least two ways to develop probability theory. The more familiar path is to treat it as its own discipline, and work from intuitive examples such as coin flips and conundrums such as the Monty Hall problem. An alternative is to first develop measure theory and analysis, and then add interpretation. Bhattacharya and Waymire take the second path. To illustrate the authors' frame of reference, consider the two definitions they give of conditional expectation. The first is as a projection of L
2 spaces. The authors rely on the reader to be familiar with Hilbert space operators and at a glance, the connection to probability may not be not apparent. Subsequently, there is a discusssion of Bayes's rule and other relevant probabilistic concepts that lead to a definition of conditional expectation as an adjustment of random outcomes from a finer to a coarser information set.
Average customer rating:
|
Brownian Motion and Index Formulas for the De Rham Complex (Mathematical Research (Vch Pub))
Kazuaki Taira
Manufacturer: Wiley-VCH Verlag GmbH
ProductGroup: Book
Binding: Paperback
General
| Science
| Subjects
| Books
Calculus
| Pure Mathematics
| Mathematics
| Science
| Subjects
| Books
Discrete Mathematics
| Pure Mathematics
| Mathematics
| Science
| Subjects
| Books
General
| Mathematics
| Science
| Subjects
| Books
Probability & Statistics
| Applied
| Mathematics
| Science
| Subjects
| Books
General
| Physics
| Science
| Subjects
| Books
Mathematical Physics
| Physics
| Science
| Subjects
| Books
Statistics
| Applied
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
Discrete Mathematics
| Pure Mathematics
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
General
| Physics
| Professional Science
| Professional & Technical
| Subjects
| Books
Mathematical Physics
| Physics
| Professional Science
| Professional & Technical
| Subjects
| Books
ASIN: 3527401393 |
Book Description
This book is an easy-to-read reference providing a link between partial differential equations (pde), stochastic analysis, and index theory. Most mathematicians working in pde are only vaguely familiar with the powerful ideas of stochastic analysis. On the other hand, the additional intuition which Taira´s book conveys might provide better insight and be helpful for their work.
In addition, the book provides a nice compendium for a large variety of facts from differential geometry, functional analysis, pseudodifferential operators, and Markov processes - for quickly looking up a theorem.
Average customer rating:
|
Probability Theory III: Stochastic Calculus (Encyclopaedia of Mathematical Sciences)
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover
General
| Science
| Subjects
| Books
Calculus
| Pure Mathematics
| Mathematics
| Science
| Subjects
| Books
General
| Mathematics
| Science
| Subjects
| Books
Probability & Statistics
| Applied
| Mathematics
| Science
| Subjects
| Books
Statistics
| Applied
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
Stochastic Modeling
| Applied
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
General
| Finance
| Accounting & Finance
| Professional & Technical
| Subjects
| Books
General
| Electronics
| Electrical & Electronics
| Engineering
| Professional & Technical
| Subjects
| Books
General
| Arts & Photography
| Subjects
| Books
All Amazon Upgrade
| Amazon Upgrade
| Stores
| Books
Arts & Photography
| Amazon Upgrade
| Stores
| Books
Engineering
| Amazon Upgrade
| Stores
| Books
Professional & Technical
| Amazon Upgrade
| Stores
| Books
Science
| Amazon Upgrade
| Stores
| Books
All Titles
| Qualifying Textbooks - Fall 2007
| Stores
| Books
Arts & Photography
| Qualifying Textbooks - Fall 2007
| Stores
| Books
Professional
| Qualifying Textbooks - Fall 2007
| Stores
| Books
Science
| Qualifying Textbooks - Fall 2007
| Stores
| Books
ASIN: 3540546871 |
Book Description
This volume of the Encyclopaedia is a survey of stochastic calculus which has become an increasingly important part of probability. The topics covered include Brownian motion, the Ito integral, stochastic differential equations and Malliavin calculus, the general theory of random processes and martingale theory. The five authors are well-known experts in the field. The first chapter of the book is an introduction which treats Brownian motion and describes the developments which lead to the definition of Ito's integral. The book addresses graduate students and researchers in probability theory and mathematical statistics and will also be used by physicists and engineers who need to apply stochastic methods.
Average customer rating:
|
Seminaire de Probabilites XXXV (Lecture Notes in Mathematics / Séminaire de Probabilités)
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback
General
| Science
| Subjects
| Books
General
| Mathematics
| Science
| Subjects
| Books
Probability & Statistics
| Applied
| Mathematics
| Science
| Subjects
| Books
Statistics
| Applied
| Mathematics
| Professional Science
| Professional & Technical
| Subjects
| Books
General
| Finance
| Accounting & Finance
| Professional & Technical
| Subjects
| Books
French
| Foreign Language Nonfiction
| Nonfiction
| Subjects
| Books
Nonfiction
| French
| Foreign Language Books
| Specialty Stores
| Books
Professional & Technical
| French
| Foreign Language Books
| Specialty Stores
| Books
All French Books
| French
| Foreign Language Books
| Specialty Stores
| Books
All Amazon Upgrade
| Amazon Upgrade
| Stores
| Books
Nonfiction
| Amazon Upgrade
| Stores
| Books
Professional & Technical
| Amazon Upgrade
| Stores
| Books
Science
| Amazon Upgrade
| Stores
| Books
All Titles
| Qualifying Textbooks - Fall 2007
| Stores
| Books
Nonfiction
| Qualifying Textbooks - Fall 2007
| Stores
| Books
Professional
| Qualifying Textbooks - Fall 2007
| Stores
| Books
Science
| Qualifying Textbooks - Fall 2007
| Stores
| Books
ASIN: 3540416595 |
Book Description
Researchers and graduate students in the theory of stochastic processes will find in this 35th volume some thirty articles on martingale theory, martingales and finance, analytical inequalities and semigroups, stochastic differential equations, functionals of Brownian motion and of Lévy processes. Ledoux's article contains a self-contained introduction to the use of semigroups in spectral gaps and logarithmic Sobolev inequalities; the contribution by Emery and Schachermayer includes an exposition for probabilists of Vershik's theory of backward discrete filtrations.
Book Description
Fractional Brownian motion (fBm) has been widely used to model a number of phenomena in diverse fields from biology to finance. This huge range of potential applications makes fBm an interesting object of study.
fBm represents a natural one-parameter extension of classical Brownian motion therefore it is natural to ask if a stochastic calculus for fBm can be developed. This is not obvious, since fBm is neither a semimartingale (except when H = ½), nor a Markov process so the classical mathematical machineries for stochastic calculus are not available in the fBm case.
Several approaches have been used to develop the concept of stochastic calculus for fBm. The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches.
Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices.
This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance. Aspects of the book will also be useful in other fields where fBm can be used as a model for applications.
Books:
- Business Dynamics: Systems Thinking and Modeling for a Complex World with CD-ROM
- Business Dynamics: Systems Thinking and Modeling for a Complex World with CD-ROM
- Castle: Medieval Days and Knights (A Sabuda & Reinhart Pop-up Book)
- CliffsNotes on Conrad's Heart of Darkness & The Secret Sharer
- College Accounting: A Practical Approach Chapters 1-12 with Study Guide and Working Papers (10th Edition)
- Consumed: How Markets Corrupt Children, Infantilize Adults, and Swallow Citizens Whole
- Covered Calls and LEAPS--A Wealth Option + DVD: A Guide for Generating Extraordinary Monthly Income (Wiley Trading)
- Crash Proof: How to Profit From the Coming Economic Collapse (Lynn Sonberg Books)
- Creating Mixed Model Value Streams: Practical Lean Techniques for Building to Demand
- Dancing With Giants: China, India, And the Global Economy
Books Index
Books Home
Recommended Books
- The Challenge of Crime: Rethinking Our Response
- Love's Unending Legacy/Love's Unfolding Dream/Love Takes Wing/Love Finds a Home
- Genetics, Evolution & Biodiversity: Nelson Advanced Science
- Genomic Imprinting and Uniparental Disomy in Medicine: Clinical and Molecular Aspects
- Home Ground: Language for an American Landscape
- StrengthsFinder 2.0: A New and Upgraded Edition of the Online Test from Gallup's Now, Discover Your
- Nobody Loves a Centurion
- Underworld: Sites of Concealment
- Landscape Ecology Principles in Landscape Architecture and Land-Use Planning
- McNally's Chance: An Archy McNally Novel by Vincent Lardo
Stochastic Differential Equations: An Introduction with Applications (Universitext)
Stochastic Calculus for Finance II: Continuous-Time Models (Springer Finance)
Methods of Mathematical Finance
Stochastic Calculus for Finance I: The Binomial Asset Pricing Model (Springer Finance)
Diffusions, Markov Processes, and Martingales (Cambridge Mathematical Library)
Heavy-Tail Phenomena: Probabilistic and Statistical Modeling (Springer Series in Operations Research and Financial Engineering)
A Modern Introduction to Probability and Statistics: Understanding Why and How (Springer Texts in Statistics)
Applied Stochastic Processes (Universitext)
