On Radon-Nikodym derivatives of finitely-additive measures induced by nonlinear transformations on hilbert space

Abstract

THE STUDY of finitely-additive probability measures arises in problems of modelling and estimation of stochastic signals with bounded variation paths in L,[O, T]. This is because physical signals are of bounded variation and of finite energy for which L,[O, T] is the natural setting and results obtained via Ito theory hold only on a set of Wiener measure 1. It is well known that integrals of paths in &JO, T] lie in a set of Wiener measure zero. Hence, there is a need to construct a theory of white noise to model large bandwidth noise arising in signal analysis with the usual properties associated with it. In a series of papers [l-3], in which he advocated the use of such a framework, Balakrishnan showed that it could indeed be done and, thus, laid the basis for a such a theory. He considered white noise to be the identity map on a Hilbert space H with standard Gauss measure thereon. It is well known that such a measure is only finitely additive on the algebra of cylinder sets and cannot be extended to the Bore1 sets of H. This line of work culminated in the excellent treatise by Kallianpur and Karandikar [4] devoted to filtering and smoothing problems. The key assumption in the development of the theory in the nonlinear context so far, has been that the signal process is assumed to be defined on a countably-additive probability space with paths in H while the measurement noise process is defined on a cylindrical probability space associated with the Gauss measure. The formulation does not allow for signal-noise dependence and one is forced to work with a quasi-cylindrical probability space (a product space). Despite the fact that the mathematical difficulties are daunting, there nevertheless arises the need to develop a complete theory of white noise in order to study modelling issues as well as signal-noise dependence, since these issues arise quite naturally in physical problems. This paper presents one step in such a direction and is motivated by the issue of likelihood ratio evaluation for signals arising in differential systems driven by white noise. The specific problem addressed in this paper is the study of Radon-Nikodym derivatives (and their evaluation) of cylindrical (or finitely-additive) measures induced by nonlinear transformations on H with standard Gauss measure thereon. In the linear case this problem was solved by Balakrishnan [5]. Balakrishnan [2] also obtained some results in the nonlinear case when the transformation is given by I + K where K defines a homogeneous, finite “Volterra” polynomial by exploiting the connection with the pioneering work of Cameron and Martin [6].