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 $Z_n:C[0,t]\to {\mathbb{R}}^n$ by $$Z_n\left(x\right)=\left({\int }_0^{t_1}h\right(s\left)dx\right(s),\dots ,{\int }_0^{t_n}h(s\left)dx\right(s\left)\right),$$ where $0<t_1<\cdots <t_n=t$ is a partition of $[0,t]$ and $h\in L_2[0,t]$ with $h\ne 0$ almost everywhere. Using a simple formula for a generalized conditional Wiener integral on $C[0,t]$ with the conditioning function $Z_n$, we evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $$G\left(x\right)=f\left(\right(e,x\left)\right)\varphi \left(\right(e,x\left)\right)$$ for $x\in C[0,t]$, where $f\in L_p\left(\mathbb{R}\right)(1\le p\le \infty )$, $e$ is a unit element in $L_2[0,t]$, and $\varphi$ is the Fourier transform of a measure of bounded variation over $\mathbb{R}$. 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 $L_2[0,t]$ used in the transformation is independent of $e$.