EVALUATION OF SOME CONDITIONAL WIENER INTEGRALS

Abstract

J. Yeh has recently introduced the concept of conditional Wiener integrals which are meant specifically the conditional expectation E (Z vertical bar X) of a real or complex valued Wiener integrable functional Z conditioned by the Wiener measurable functional X on the Wiener measure space (A precise definition of the conditional Wiener integral and a brief discussion of the Wiener measure space are given in Section 2). In [3] and [4] he derived some inversion formulae for conditional Wiener integrals and evaluated some conditional Wiener integrals E (Z vertical bar X) conditioned by X(x)=x(t) for a fixed t>0 and x in Wiener space. Thus E (Z vertical bar X) is a real or complex valued function on R. In this paper we shall be concerned with the random vector X given by X(x) = (x(s),..,x(s )) for every x in Wiener space where 0=s =t. In Section 3 we will evaluate some conditional Wiener integrals E (Z vertical bar X) which are real or complex valued functions on the n-dimensional Euclidean space R . Thus we extend Yeh's results [4] for the random variable X given by X(x)=x(t) to the random vector X given by X(x)=(x(s).., x(s )).