Parameter Estimation in Stochastic Differential Equations
Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modelling complex phenomena and making beautiful decisions. The subject has attracted researchers from several ar
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1923
Jaya P.N. Bishwal
Parameter Estimation in Stochastic Differential Equations
ABC
Author Jaya P.N. Bishwal Department of Mathematics and Statistics University of North Carolina at Charlotte 376 Fretwell Bldg. 9201 University City Blvd. Charlotte NC 28223-0001 USA e-mail: [email protected] URL: http://www.math.uncc.edu/∼jpbishwa
Library of Congress Control Number: 2007933500 Mathematics Subject Classification (2000): 60H10, 60H15, 60J60, 62M05, 62M09 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN 978-3-540-74447-4 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-74448-1 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. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2008 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 by the authors and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
SPIN: 12111664
41/SPi
543210
To the memory of my late grand father and to my parents and brothers for their love and affection
Preface
I am indebted to my advisor Arup Bose from who I learned inference for diffusion processes during my graduate studies. I have benefited a lot from my discussions with Yacine Ait-Sahalia, Michael Sørensen, Yuri Kutoyants and Dan Crisan. I am grateful to all of them for their advice.
Charlotte, NC January 30, 2007
Jaya P.N. Bishwal
Contents
Basic Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .XIII 1
Parametric Stochastic Differential Equations . . . . . . . . . . . . . . .
1
Part I Continuous Sampling 2
3
4
Rates of Weak Convergence of Estimators in Homogeneous Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Berry-Esseen Bounds for Estimators in the OrnsteinUhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Rates of Convergence in the Bernstein-von Mises Theorem for Ergodic Diffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Rates of Convergence of the Posterior Distrib
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