Stochastic Calculus with Infinitesimals
Stochastic analysis is not only a thriving area of pure mathematics with intriguing connections to partial differential equations and differential geometry. It also has numerous applications in the natural and social sciences (for instance in financial ma
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2067
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Frederik S. Herzberg
Stochastic Calculus with Infinitesimals
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Frederik S. Herzberg Bielefeld University Institute of Mathematical Economics Bielefeld, Germany and Ludwig Maximilian University of Munich Munich Center for Mathematical Philosophy Munich, Germany
ISBN 978-3-642-33148-0 ISBN 978-3-642-33149-7 (eBook) DOI 10.1007/978-3-642-33149-7 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012951626 Mathematics Subject Classification (2010): 03H05; 60G05; 91B25; 81Q30; 60G51; 60H05; 60H10 c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to Professor Edward Nelson on the occasion of his eightieth birthday in 2012
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Abstract
This short monograph develops basic stochastic analysis—including Itˆo’s formula, Girsanov’s theorem, the Feynman–Kac formula, and results about L´evy processes with finite-variation jump part—and select applications in the framework of Edward Nelson’s Radically elementary probability theory [Annals of Mathematics Studies, 117, Princeton, NJ: Princeton University Press, 1987]. This approac
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