Transformation of Measure on Wiener Space (Springer Monographs in Mathematics)

This book gives a systematic presentation of the main results on the transformation of measure induced by shift transformations on Wiener space. This topic has its origins in the work of Cameron and Martin (anticipative shifts, 1940's) and that of Girsanov (non-anticipative shifts, 1960's). It played an important role in the development of non-anticipative stochastic calculus and itself developed under the impulse of the stochastic calculus of variations. The recent results presented in the book include a dimension-free form of the Girsanov theorem, the transformations of measure induced by anticipative non-invertible shift transformations, the transformation of measure induced by flows, the extension of the notions of Sard lemma and degree theory to Wiener space, generalized distribution valued Radon-Nikodym theorems and measure preserving transformations. Basic probability theory and the Ito calculus are assumed known; the necessary results from the Malliavin calculus are presented in the appendix. Aimed at graduate students and researchers, it can be used as a text for a course or a seminar. Front Matter....Pages I-XIII Introduction....Pages 1-4 Some Background Material and Preliminary Results....Pages 5-19 Transformation of Measure Induced by Adapted Shifts....Pages 21-51 Transformation of Measure Induced by General Shifts....Pages 53-98 The Sard Inequality....Pages 99-113 Transformation of Measure Under Anticipative Flows....Pages 115-156 Monotone Shifts....Pages 157-180 Generalized Radon-Nikodym Derivatives....Pages 181-205 Random Rotations....Pages 207-231 The Degree Theorem on Wiener Space....Pages 233-254 Back Matter....Pages 255-297 This unique book on the subject addresses fundamental problems and will be the standard reference for a long time to come. The authors have different scientific origins and combine these successfully, creating a text aimed at graduate students and researchers that can be used for courses and seminars.