Transformation of Measure Induced by Adapted Shifts
In this chapter we consider the following setup: (Ω,F, (F t , t ∈ [0, 1]), Θ) or, in short (Ω, F,.F.,Θ) is a filtered probability space, i.e., (F t , t ∈ [0, 1]) is a right continuous, increasing family of sub-sigma fields of. F. We suppose that F 0 conta
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Springer-Verlag Berlin Heidelberg GmbH
A. Süleyman Üstünel • Moshe Zakai
Transformation of Measure on Wiener Space
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Springer
Ali Siileyman Ostiinel Ecole Nationale Superieure des TeIecommunications Departement Infres 46, rue Barrault F-75013 Paris, France e-mail: [email protected] MosheZakai Technion -Israel Institute of Technology Faculty of Electrical Engineering 32000 Haifa, Israel e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ostiinel, Ali Siileyman: Transformation of measure on Wiener space I Ali Siileyman Ostiinel; Moshe Zakai. (Springer monographs in mathematics) ISBN 978-3-642-08572-7 ISBN 978-3-662-13225-8 (eBook) DOI 10.1007/978-3-662-13225-8
Mathematics Subject Classification (1991): 60H07, 60H05, 60H25, 60G15, 60G30, 60G35, 62CI0, 26A16, 28C20, 46G12, 47H05, 47Hl
ISBN 978-3-642-08572-7
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Preface
The notion of transformation of measure is of fundamental importance in Analysis and Probability Theory. In the context of Ite. calculus, the Girsanov theorem elassified the structure of the Random-Nikodym derivative and turned out to be of fundamental importance for the development of Itö Calculus (its extension to martingales, the martingale problem, weak solutions of stochastic differential equations) and its applications (e.g. filtering, stochastic control, and mathematical finance). This set up is associated with a time ftow and cannot be extended to cases such as stochastic partial differential equations on subsets of IRn , n > 1, wh ich lack the time ftow structure. The problem of transformation of measures for general structures started in the late forties with the work of Cameron and Martin. The work of Ramer in the mid-seventies and
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