## Annals of Functional Analysis

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

Dong Hyun Cho

#### 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 $\begin{eqnarray*}Z_{n}(x)=(\int_{0}^{t_{1}}h(s)\,dx(s),\ldots,\int_{0}^{t_{n}}h(s)\,dx(s)),\end{eqnarray*}$ where $0\lt t_{1}\lt \cdots\lt t_{n}=t$ is a partition of $[0,t]$ and $h\inL_{2}[0,t]$ with $h\neq0$ 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 $\begin{eqnarray*}G(x)=f((e,x))\phi((e,x))\end{eqnarray*}$ for $x\in C[0,t]$, where $f\in L_{p}(\mathbb{R})(1\le p\le\infty)$, $e$ is a unit element in $L_{2}[0,t]$, and $\phi$ 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 $e$.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 2 (2016), 358-370.

Dates
Accepted: 2 November 2015
First available in Project Euclid: 8 April 2016

https://projecteuclid.org/euclid.afa/1460141561

Digital Object Identifier
doi:10.1215/20088752-3544830

Mathematical Reviews number (MathSciNet)
MR3484389

Zentralblatt MATH identifier
1346.46038

#### Citation

Cho, Dong Hyun. Scale transformations for present position-dependent conditional expectations over continuous paths. Ann. Funct. Anal. 7 (2016), no. 2, 358--370. doi:10.1215/20088752-3544830. https://projecteuclid.org/euclid.afa/1460141561

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