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An Introduction to Stochastic Differential Equations
This book provides a quick, but very readable introduction to stochastic differential equations, that is, to differential equations subject to additive "white noise" and related random disturbances. The exposition is concise and strongly focused upon the interplay between probabilistic intuition and mathematical rigor. Topics include a quick survey of measure theoretic probability theory, followed by an introduction to Brownian motion and the Îto stochastic calculus, and finally the theory of stochastic differential equations. The text also includes applications to partial differential equations, optimal stopping problems and option pricing.
This book can be used as a text for senior undergraduates or beginning graduate students in mathematics, applied mathematics, physics, financial mathematics, etc., who want to learn the basics of stochastic differential equations. The reader is assumed to be fairly familiar with measure theoretic mathematical analysis, but is not assumed to have any particular knowledge of probability theory (which is rapidly developed in Chapter 2 of the book)
This book can be used as a text for senior undergraduates or beginning graduate students in mathematics, applied mathematics, physics, financial mathematics, etc., who want to learn the basics of stochastic differential equations. The reader is assumed to be fairly familiar with measure theoretic mathematical analysis, but is not assumed to have any particular knowledge of probability theory (which is rapidly developed in Chapter 2 of the book)
- GenresMathematics
152 pages, Paperback
First published November 1, 2013
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Displaying 1 - 2 of 2 reviews
December 8, 2014
This book is an outstanding introduction to this subject, focusing on the Ito calculus for stochastic differential equations (SDEs). For anyone who is interested in mathematical finance, especially the Black-Scholes-Merton equation for option pricing, this book contains sufficient detail to understand the provenance of this result and its limitations.
I recommend this book to advanced undergraduate mathematics students and graduate finance students. It is ideal as a primer prior to reading, for instance, Bernt Oksendal's "Stochastic Differential Equations: An Introduction with Applications", John S. Cochrane's "Asset Pricing", or Kiyoshi Ito's "Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces".
An interesting feature at the end of the book is a brief exposition on the Stratonovich calculus, which is an alternative to the Ito calculus that has some excellent stability characteristics, especially if the underlying variance (or volatility, in financial problems) is not strictly governed by a Gaussian "white noise" perturbation.
I recommend this book to advanced undergraduate mathematics students and graduate finance students. It is ideal as a primer prior to reading, for instance, Bernt Oksendal's "Stochastic Differential Equations: An Introduction with Applications", John S. Cochrane's "Asset Pricing", or Kiyoshi Ito's "Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces".
An interesting feature at the end of the book is a brief exposition on the Stratonovich calculus, which is an alternative to the Ito calculus that has some excellent stability characteristics, especially if the underlying variance (or volatility, in financial problems) is not strictly governed by a Gaussian "white noise" perturbation.
May 22, 2024
A great introductory textbook on the Itô stochastic calculus and stochastic ordinary differential equations. This book is especially suitable for readers with a background in measure theory without any extensive knowledge of probability, which is developed in the second chapter. Subsequent chapters deal with the theory of Brownian motion and the Itô's stochastic integral (the Stratonovich integral is briefly discussed at the end of the book), which are vital for the main topic of the book - stochastic ordinary differential equations. The basic theory thereof is provided in the fifth chapter with the focus on the existence and uniqueness of solutions. Some methods for solving linear stochastic ordinary differential equations are illustrated as well. The book is concluded with applications of stochastic calculus (if you are a PDE-oriented mathematician, you will definitely appreciate the subchapter on the applications to PDEs and the Feynman-Kac formula).
As the author himself writes in the preface, his aim is merely to survey the basics of stochastic calculus, meaning that some of the more challenging proofs have been omitted and most of the other proofs are done only for special simple examples, such as the step functions with a reference to the density/approximation argument for the general case. Therefore, if you are looking for a more rigorous textbook, you might want to use a more advanced textbook instead (with Evans possible serving as a complement).
As the author himself writes in the preface, his aim is merely to survey the basics of stochastic calculus, meaning that some of the more challenging proofs have been omitted and most of the other proofs are done only for special simple examples, such as the step functions with a reference to the density/approximation argument for the general case. Therefore, if you are looking for a more rigorous textbook, you might want to use a more advanced textbook instead (with Evans possible serving as a complement).
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