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Stochastic Calculus: Applications in Science and Engineering, by Mircea Grigoriu
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Algebraic, differential, and integral equations are used in the applied sciences, en gineering, economics, and the social sciences to characterize the current state of a physical, economic, or social system and forecast its evolution in time. Generally, the coefficients of and/or the input to these equations are not precisely known be cause of insufficient information, limited understanding of some underlying phe nomena, and inherent randonmess. For example, the orientation of the atomic lattice in the grains of a polycrystal varies randomly from grain to grain, the spa tial distribution of a phase of a composite material is not known precisely for a particular specimen, bone properties needed to develop reliable artificial joints vary significantly with individual and age, forces acting on a plane from takeoff to landing depend in a complex manner on the environmental conditions and flight pattern, and stock prices and their evolution in time depend on a large number of factors that cannot be described by deterministic models. Problems that can be defined by algebraic, differential, and integral equations with random coefficients and/or input are referred to as stochastic problems. The main objective of this book is the solution of stochastic problems, that is, the determination of the probability law, moments, and/or other probabilistic properties of the state of a physical, economic, or social system. It is assumed that the operators and inputs defining a stochastic problem are specified.
- Sales Rank: #2336201 in Books
- Published on: 2002-09-24
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x 1.69" w x 6.14" l, 2.76 pounds
- Binding: Hardcover
- 792 pages
Review
"When familiar at least with the basics of measure theoretic probability, one may use this book to get a feel for the type of problems one tackles with the given methods and the type of results one can expect and then proceed to more detailed expositions.... The book is an impressive achievement." ―Mathematical Reviews
"The large number of examples...as well as the good survey on advanced mathematical techniques make the textbook very valuable for people working in mathematics with a view towards applications.... The applications range from climate dynamics over material sciences and mechanics to pattern formation, physics and finance. The presentation is very successful in providing a sound mathematical background without getting lost in too much technical detail. Most results are proved completely, for the others the main ideas and intuition are given and references for detailed expositions are given.... At the end of each chapter a list of instructive exercises is provided. The long list of references covers theoretical mathematical works as well as highly applied areas. Altogether this textbook has all the necessary prerequisites for boosting the interdisciplinary collaboration between applied scientists and probabilists." ―Zentralblatt Math
"This is an impressive compendium, which, on its first 280 pages, provides a quite thorough review of probability theory, stochastic processes, Itô’s formula and stochastic differential equations.... Although the focus is on the mathematical theory, many examples from sciences or engineering illustrate the concepts and methods. Each chapter ends with a number of problems. This book will be of substantial interest to the applied scientists seeking methods to solve stochastic problems." ―Monatshefte für Mathematik
From the Back Cover
This work focuses on analyzing and presenting solutions for a wide range of stochastic problems that are encountered in applied mathematics, probability, physics, engineering, finance, and economics. The approach used reduces the gap between the mathematical and engineering literature. Stochastic problems are defined by algebraic, differential or integral equations with random coefficients and/or input. However, it is the type, rather than the particular field of application, that is used to categorize these problems.
An introductory chapter outlines the types of stochastic problems under consideration in this book and illustrates some of their applications. A user friendly, systematic exposition unfolds as follows: The essentials of probability theory, random processes, stochastic integration, and Monte Carlo simulation are developed in chapters 2--5. The Monte Carlo method is used extensively to illustrate difficult theoretical concepts and solve numerically some of the stochastic problems in chapters 6--9.
Key features:
* Computational skills developed as needed to solve realistic stochastic problems
* Classical mathematical notation used, and essential theoretical facts boxed
* Numerous examples from applied sciences and engineering
* Complete proofs given; if too technical, notes clarify the idea and/or main steps
* Problems at the end of each chapter reinforce applications; hints given.
* Good bibliography at the end of every chapter
* Comprehensive index
This work is unique, self-contained, and far from a collection of facts and formulas. The analytical and numerical methods approach for solving stochastic problems may be used for self-study by a variety of researchers, and in the classroom by first year graduate students.
Most helpful customer reviews
12 of 13 people found the following review helpful.
Very dense. Proof-oriented. Excellent use of measure theory.
By Alexander C. Zorach
This book is a lot denser than the pretty picture on the cover and the presence of occasional graphs would lead you to expect. It's like 2-3 difficult graduate courses and a bunch of more advanced material for reference all crammed into one place.
The exercises are mostly proof-oriented and would be good preparation for someone looking to do research in this field. I find this book doesn't exactly fit its subtitle of "Applications in Science and Engineering". It is primarily a mathematics book that acknowledges and sometimes discusses applications as motivation for the material. I think it would be more useful to a mathematician than to an engineer or scientist.
My favorite aspect of the book is its no-nonsense treatment of measure-theoretic probability. The book presents the bare minimum necessary for its use, and leaves out some inessential proofs (giving references). I think this is a great way to learn measure theory: first develop an intuitive sense of it and an appreciation of its usefulness, and then you can go back and learn it rigorously from another book. The book provides enough background that you don't need to look elsewhere in order to understand the measure theory used in the text.
I've found a few mistakes in the book, which I find annoying, because I would not trust myself to catch any mistakes beyond maybe chapter 3.
Overall though, I think this is an excellent book. It is about as clear as you can get on the material it covers without watering it down. Lastly, this book covers so much material, most people will not be able to exhaust its contents, and will find it useful as a reference.
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