Introduction to Stochastic Calculus for Finance A New Didactic Appro
The large number of already available textbooks on stochastic calculus with specific applications to finance requires a justification for another contribution to this subject. The justifcation is mainly pedagogical. These lecture notes start with an eleme
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Founding Editors: M. Beckmann H.P. Künzi Managing Editors: Prof. Dr. G. Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr. 140/AVZ II, 58084 Hagen, Germany Prof. Dr. W. Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr. 25, 33615 Bielefeld, Germany Editorial Board: A. Basile, A. Drexl, H. Dawid, K. Inderfurth, W. Kürsten, U. Schittko
Dieter Sondermann
Introduction to Stochastic Calculus for Finance A New Didactic Approach
With 6 Figures
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Prof. Dr. Dieter Sondermann Department of Economics University of Bonn Adenauer Allee 24 53113 Bonn, Germany E-mail: [email protected]
ISBN-10 3-540-34836-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-34836-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, 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. Typesetting: Camera ready by author Cover: Erich Kirchner, Heidelberg Production: LE-TEX, Jelonek, Schmidt & Vöckler GbR, Leipzig SPIN 11769675
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To Freddy, Hans and Marek, who patiently helped me to a deeper understanding of stochastic calculus.
Preface
There are by now numerous excellent books available on stochastic calculus with specific applications to finance, such as Duffie (2001), ElliottKopp (1999), Karatzas-Shreve (1998), Lamberton-Lapeyre (1995), and Shiryaev (1999) on different levels of mathematical sophistication. What justifies another contribution to this subject? The motivation is mainly pedagogical. These notes start with an elementary approach to continuous time methods of Itˆ o’s calculus due to F¨ollmer. In an fundamental, but not well-known paper published in French in the Seminaire de Probabilit´e in 1981 (see Foellmer (1981)), F¨ollmer showed that one can develop Itˆo’s calculus without probabilities as an exercise in real analysis. 1 The notes are based on courses offered regularly to graduate students in economics and mathematics at the University of Bonn choosing “financial economics” as special topic. To students interested in finance the course opens a quick (but by no means “dirty”) road to
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