Introduction to Stochastic Integration

by Chung, K. L.

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Title Statements

Full title :Introduction to Stochastic Integration / by K.L. Chung, R.J. Williams.

Summary

A highly readable introduction to stochastic integration and stochastic differential equations, this book combines developments of the basic theory with applications. It is written in a style suitable for the text of a graduate course in stochastic calculus, following a course in probability. Using the modern approach, the stochastic integral is defined for predictable integrands and local martingales; then Its change of variable formula is developed for continuous martingales. Applications include a characterization of Brownian motion, Hermite polynomials of martingales, the Feynman-Kac functional and Schrdinger equation. For Brownian motion, the topics of local time, reflected Brownian motion, and time change are discussed. New to the second edition are a discussion of the Cameron-Martin-Girsanov transformation and a final chapter which provides an introduction to stochastic differential equations, as well as many exercises for classroom use. This book will be a valuable resource to all mathematicians, statisticians, economists, and engineers employing the modern tools of stochastic analysis. The text also proves that stochastic integration has made an important impact on mathematical progress over the last decades and that stochastic calculus has become one of the most powerful tools in modern probability theory. Journal of the American Statistical Association An attractive textwritten in [a] lean and precise styleeminently readable. Especially pleasant are the care and attention devoted to details A very fine book. Mathematical Reviews.

Edition Statements

Publication Date:1990.
Edition:Second edition

Publication Statements

Publisher(s):Birkhäuser Boston
Place of Publication:Boston, MA
Language:eng
Publication Date:1990.

The Physical Object

Description:1 online resource (300 pages)

Edition Identifiers

ISBN:9781461244806, 1461244803, 9781461288374, 1461288371, 1461495873, 9781461495871
DOI:10.1007/978-1-4612-4480-6
oclc:ocn840279690
LC Classification:QA273.A1-274.9
Dewey Decimal:519.2

1. Preliminaries -- 1.1 Notations and Conventions -- 1.2 Measurability, LP Spaces and Monotone Class Theorems -- 1.3 Functions of Bounded Variation and Stieltjes Integrals -- 1.4 Probability Space, Random Variables, Filtration -- 1.5 Convergence, Conditioning -- 1.6 Stochastic Processes -- 1.7 Optional Times -- 1.8 Two Canonical Processes -- 1.9 Martingales -- 1.10 Local Martingales -- 1.11 Exercises -- 2. Definition of the Stochastic Integral -- 2.1 Introduction -- 2.2 Predictable Sets and Processes -- 2.3 Stochastic Intervals -- 2.4 Measure on the Predictable Sets -- 2.5 Definition of the Stochastic Integral -- 2.6 Extension to Local Integrators and Integrands -- 2.7 Substitution Formula -- 2.8 A Sufficient Condition for Extendability of?z -- 2.9 Exercises -- 3. Extension of the Predictable Integrands -- 3.1 Introduction -- 3.2 Relationship between P, O, and Adapted Processes -- 3.3 Extension of the Integrands -- 3.4 A Historical Note -- 3.5 Exercises -- 4. Quadratic Variation Process -- 4.1 Introduction -- 4.2 Definition and Characterization of Quadratic Variation -- 4.3 Properties of Quadratic Variation for an L2-martingale -- 4.4 Direct Definition of?M -- 4.5 Decomposition of…