Stochastic Calculus Applications in Science and Engineering
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 mathemati
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Mircea Grigoriu
Stochastic Calculus Applications in Science and Engineering
Springer Science+Business Media, LLC
Mircea Grigoriu Cornell University School of Civil and Environmental Engineering Ithaca, NY 14853 U.S.A.
Library of Congress Cataloging-in-Publication Data Grigoriu, Mircea. Stochastic calculus : applications in science and engineering I Mircea Grigoriu. p.cm. Includes bibliographical references and index.
ISBN 978-1-4612-6501-6
1. Stochastic analysis. I. Title.
QA274.2.G75 2002 519.2-dc21
2002074386 CIP
ISBN 978-1-4612-6501-6 ISBN 978-0-8176-8228-6 (eBook) DOI 10.1007/978-0-8176-8228-6 AMS Subject Oassifications: 37A50, 60GXX, 35SXX, 35QXX
Printed on acid-free paper. © 2002 Springer Science+Business Media New York Originally published by Birkhiiuser Boston in 2002 Softcover reprint of the hardcover 1st edition 2002
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All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden . The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. SPIN 10835902
Typeset by the author.
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Contents 1 Introduction
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2 Probability Theory 2.1 Introduction . . 2.2 Probability space .. 2.2.1 Sample space . 2.2.2 a-field 2.2.3 Probability measure 2.3 Construction of probability spaces 2.3.1 Countable sample space 2.3.2 Product probability space . 2.3.3 Extension of probability measure 2.3.4 Conditional probability . 2.3.5 Sequence of sets 2.3.6 Sequence of events 2.4 Measurable functions . . . 2.4.1 Properties . . . . . 2.4.2 Definition of random variable 2.4.3 Measurable transformations 2.5 Integration and expectation . . . . . . 2.5.1 Expectation operator . . . . . 2.5.1.1 Finite-valued simple random variables 2.5.1.2 Positive random variables . . . . 2.5.1.3 Arbitrary random variables .... 2.5.2 Properties of integrals of random variables 2.5.2.1 Finite number of random variables 2.5.2.2 Sequence of random variables . 2.5.2.3 Expectation ............ 2.6 The Lq (Q, F, P) space . . . . . . . . . . . . . . . . . . . . 2.7 Independence . . . . . . . . . . . 2.7.1 Independence of a-fields . 2.7.2 Independence of events .
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2.7.3 Independence of random variables 2.8 The Fubini theorem . . . . 2.9 Radon-Nikodym derivative . 2.10 Random variables . . . . . . 2.10.1 Distribution function 2.10.2 Density function . . 2.10.3 Characteristic function 2.10.3.1 Propert
