Introduction

Abstract

In this book we study the properties of the image of Wiener measure under several (point) transformations of the flat Wiener space. One of the main topics dealt with concerns the absolute continuity and the calculation of Radon-Nikodym derivatives associated with these transformations. This subject has played a paramount role in the development of the basic theory and the applications of stochastic calculus and will, no doubt, continue to play an important role in future developments of stochastic analysis. Indeed, the theory of transformation of measures induced by adapted perturbations of the Wiener path (and its extension to martingales) played a key role in the construction of weak solutions to stochastic differential equations, the extension of almost-sure results for quasimartingales to semimartingales, the theories of non-linear filtering and optimal stochastic control, financial mathematics, large deviations, quantum physics, stochastic calculus of variations of Paul Malliavin —to mention just a few examples. More recently, results on the transformation of measure under general, not necessarily adapted, shifts have been applied to study the existence and Markov property of solutions to stochastic partial differential equations, nonlinear filtering in a non adapted context and oscillatory integrals.