Parameter estimation for stochastic processes

Abstract

�9 A stochastic process [x(t), t E I], or x for short, has associated with it a probability measure Px defined on suitable subsets of the space of sample functions on I. The problems of determining when measures Px and P~ associated with processes x and y are mutually absolutely continuous and of computing the Radon-Nikodym derivative dPx/dPy have been much investigated in recent years. In particular, a necessary and sufficient criterion has been given in case x and y are Gaussian for determining the mutual absolute continuity of Px and Py [3]. If we take I to be an interval and x and y to have zero means and correlation functions R~(s, t) and R~(s, t) whose associated integral operators on L~(dt, I) are compact, then the criterion is that R;�89 �89 have an extension to a Hilbert--Schmidt operator and under these circumstances dPx/dPy can be expressed in terms of the eigenfunctions and eigenvalues of this operator. In parameter estimation, however, where whole families (P~) of measures must be considered, results of this type (which tend to involve separate calculations for each pair 61 and as) often involve prohibitive amounts of calculation and also obscure the role played by the parameter itself. In [8] we attacked this problem under the assumption that the processes x~ were gotten from each other by the application of a one-parameter group T~ of transformations acting on the sample functions of the process. Specifically, we assumed given an algebra F of bounded random variables on which T~ operated as a group of automorphisms (intuitively (T~/)(x)=/(T~x)) such that the derivative DT~/(x)=~Tj(x)/~o~ existed and was uniformly bounded in ~ and x. It was shown there that the existence of a random variable ~ satisfying j q~/dP~ = S D/dPz for all / in F implied the existence of a strongly continuous one-parameter group [V(~)[~>0] of contractions on LI(P~)