2014 Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
Dong Hyun Cho
Abstr. Appl. Anal. 2014: 1-12 (2014). DOI: 10.1155/2014/916423

## 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 stochastic process $Z:C[0,T]{\times}[0,T]\to \mathbb{R}$ by $Z(x,t)={\int }_{0}^{t}\mathrm{‍}h(u)dx(u)+x(0)+a(t)$, for $x\in C[0,T]$ and $t\in [0,T]$, where $h\in {L}_{2}[0,T]$ with $h\ne 0$ a.e. and $a$ is a continuous function on $[0,T]$. Let ${Z}_{n}:C[0,T]\to {\mathbb{R}}^{n+1}$ and ${Z}_{n+1}:C[0,T]\to {\mathbb{R}}^{n+2}$ be given by ${Z}_{n}(x)=(Z(x,{t}_{0}),Z(x,{t}_{1}),\dots ,Z(x,{t}_{n}))$ and ${Z}_{n+1}(x)=(Z(x,{t}_{0}),Z(x,{t}_{1}),\dots ,Z(x,{t}_{n}),Z(x,{t}_{n+1}))$, where $0={t}_{0}<{t}_{1}<\cdots <{t}_{n}<{t}_{n+1}=T$ is a partition of $[0,T]$. In this paper we derive two simple formulas for generalized conditional Wiener integrals of functions on $C[0,T]$ with the conditioning functions ${Z}_{n}$ and ${Z}_{n+1}$ which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function $\text{e}\text{x}\text{p}\{{\int }_{0}^{T}\mathrm{‍}Z(x,t)d{m}_{L}(t)\}$ including the time integral on $C[0,T]$.

## Citation

Dong Hyun Cho. "Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space." Abstr. Appl. Anal. 2014 1 - 12, 2014. https://doi.org/10.1155/2014/916423

## Information

Published: 2014
First available in Project Euclid: 2 October 2014

zbMATH: 07023304
MathSciNet: MR3226236
Digital Object Identifier: 10.1155/2014/916423