Scale transformations for present position-dependent conditional expectations over continuous paths

Abstract

Let C[0,t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,t], and define a random vector Zn:C[0,t]→Rn by Zn(x)=(∫0t1h(s)dx(s),…,∫0tnh(s)dx(s)), where 0<t1<⋯<tn=t is a partition of [0,t] and h∈L2[0,t] with h≠0 almost everywhere. Using a simple formula for a generalized conditional Wiener integral on C[0,t] with the conditioning function Zn, we evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function G(x)=f((e,x))ϕ((e,x)) for x∈C[0,t], where f∈Lp(R)(1≤p≤∞), e is a unit element in L2[0,t], and ϕ is the Fourier transform of a measure of bounded variation over R. We then express the generalized analytic conditional Feynman integral of G as two kinds of limits of nonconditional generalized Wiener integrals with a polygonal function and cylinder functions using a change-of-scale transformation. The choice of a complete orthonormal subset of L2[0,t] used in the transformation is independent of e.