Introduction to Stochastic Integration
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 calcul
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K.L. Chung
R.l. Williams
Introduction to Stochastic Integration Second Edition
Birkhauser Boston • Basel • Berlin
K.L. Chung Department of Mathematics Stanford University Stanford, California 94305, USA
R.J. Williams Department of Mathematics University of California at San Diego La Jolla, California 92093, USA
Cover Image. A Brownian motion B starts at a point x inside the domain D and first leaves D at BT • Under conditions explained in Section 6.4, a solution to the Schrodinger equation ! t.,p + q,p = 0 in D that approaches f on the boundary of D can be represented by 2
The Brownian path used in this illustration is from The Fractal Geometry of Nature © 1982 by Benoit B. Mandelbrot and is used with his kind permission.
Library of Congress Cataloging-in-Publication Data Chung, Kai Lai, 1917Introduction to stochastic integration I K.L. Chung, R.1. Williams. - 2nd ed. p. cm. - (Probability and its applications) Includes bibliographical references (p. ) and index. ISBN·13:978-1-4612-8837·4 1. Integrals, Stochastic. 2. Martingales (Mathematics) I. Williams, R. 1. (Ruth 1.), 1955. II. Title. ill. Series. QA274.22.C48 1990 519.2-dc20 90-1020 CIP Printed on acid-free paper. ©Birkhauser Boston, 1990 Softcover reprint of the hardcover 2nd edition 1990 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use, or the internal or personal use of specific clients, is granted by Birkhauser Boston, Inc., for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to Birkhauser Boston, Inc., 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN-13:978-1-4612-8837·4 DOl: 10.1007/978-1-4612-4480·6
e-ISBN-13:978-1-4612-4480·6
Camera-ready copy provided by the authors using TEX.
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PREFACE
This is a substantial expansion of the first edition. The last chapter on stochastic differential equations is entirely new, as is the longish section §9.4 on the Cameron-Martin-Girsanov formula. Illustrative examples in Chapter 10 include the warhorses attached to the names of L. S. Ornstein, Uhlenbeck and Bessel, but also a novelty named after Black and Scholes. The Feynman-Kac-Schrooinger development (§6.4) and the material on reflected Brownian motions (§8.5) have been updated. Needless to say, there are scattered over the text minor improvements and corrections to the first edition. A Russian translation of the latter, without changes, appeared in 1987. Stochastic integration has grown in both theoretical and applicable importance in the last decade, to the extent that this new tool is now sometimes employed without heed to its rigorous requirements. This is no mor
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