E-Book Overview
The heart of the book is the development of a short-time asymptotic expansion for the heat kernel. This is explained in detail and explicit examples of some advanced calculations are given. In addition some advanced methods and extensions, including path integrals, jump diffusion and others are presented.
The book consists of four parts: Analysis, Geometry, Perturbations and Applications. The first part shortly reviews of some background material and gives an introduction to PDEs. The second part is devoted to a short introduction to various aspects of differential geometry that will be needed later. The third part and heart of the book presents a systematic development of effective methods for various approximation schemes for parabolic differential equations. The last part is devoted to applications in financial mathematics, in particular, stochastic differential equations.
Although this book is intended for advanced undergraduate or beginning graduate students in, it should also provide a useful reference for professional physicists, applied mathematicians as well as quantitative analysts with an interest in PDEs.
E-Book Content
Ivan G. Avramidi
Heat Kernel Method and its Applications
Ivan G. Avramidi
Heat Kernel Method and its Applications
Ivan G. Avramidi Department of Mathematics New Mexico Tech Socorro, New Mexico, USA
ISBN 978-3-319-26265-9 ISBN 978-3-319-26266-6 (eBook) DOI 10.1007/978-3-319-26266-6 Library of Congress Control Number: 2015956332 Mathematics Subject Classification (2010): 35-01, 35K05, 35K08, 35K10, 35K67, 35Q91, 58-01, 58J05, 58J35, 58J37, 81Q20, 91G20, 91G30, 91G80 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 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. 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. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com)
To my wife Valentina, my son Grigori, and my parents
Preface
I am a mathematical physicist. I have been working in mathematical physics over thirty years. The primary focus of my research, until recently, has been developing advanced methods of geometric analysis and applying them to quantum theory. A financial industry practitioner might ask a natural question: “Is there anything useful a mathematical physicist can tell me?” Well, I asked myself the same question when I got an email from Michel Crouhy, the head of Research and Development at NATIXIS Corporate and Investment Bank in Paris, inviting me to present a series of lectures for the members of his group. Very soon, with the help of Olivier Croissant, I realized that one of the major problems of quantitative finance, at least in option pricing theory,