Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space

Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space

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]×[0,T]→R by Z(x,t)=∫0t‍h(u)dx(u)+x(0)+a(t), for x∈C[0,T] and t∈[0,T], where h∈L2[0,T] with h≠0 a.e. and a is a continuous function on [0,T]. Let...

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Main Author: Dong Hyun Cho
Format: Article
Language:English
Published: Hindawi Limited 2014-01-01
Series:Abstract and Applied Analysis
Online Access:http://dx.doi.org/10.1155/2014/916423
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spelling doaj-3111c1a3664647d9b3c7ce300bd8df142020-11-24T21:56:00ZengHindawi LimitedAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/916423916423Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function SpaceDong Hyun Cho0Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of KoreaLet 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]×[0,T]→R by Z(x,t)=∫0t‍h(u)dx(u)+x(0)+a(t), for x∈C[0,T] and t∈[0,T], where h∈L2[0,T] with h≠0 a.e. and a is a continuous function on [0,T]. Let Zn:C[0,T]→Rn+1 and Zn+1:C[0,T]→Rn+2 be given by Zn(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn)) and Zn+1(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn),Z(x,tn+1)), where 0=t0<t1<⋯<tn<tn+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 Zn and Zn+1 which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function exp{∫0T‍Z(x,t)dmL(t)} including the time integral on C[0,T].http://dx.doi.org/10.1155/2014/916423
collection DOAJ
language English
format Article
sources DOAJ
author Dong Hyun Cho
spellingShingle Dong Hyun Cho
Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
Abstract and Applied Analysis
author_facet Dong Hyun Cho
author_sort Dong Hyun Cho
title Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
title_short Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
title_full Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
title_fullStr Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
title_full_unstemmed Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space
title_sort analogues of conditional wiener integrals with drift and initial distribution on a function space
publisher Hindawi Limited
series Abstract and Applied Analysis
issn 1085-3375
1687-0409
publishDate 2014-01-01
description 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]×[0,T]→R by Z(x,t)=∫0t‍h(u)dx(u)+x(0)+a(t), for x∈C[0,T] and t∈[0,T], where h∈L2[0,T] with h≠0 a.e. and a is a continuous function on [0,T]. Let Zn:C[0,T]→Rn+1 and Zn+1:C[0,T]→Rn+2 be given by Zn(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn)) and Zn+1(x)=(Z(x,t0),Z(x,t1),…,Z(x,tn),Z(x,tn+1)), where 0=t0<t1<⋯<tn<tn+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 Zn and Zn+1 which contain drift and initial distribution. As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function exp{∫0T‍Z(x,t)dmL(t)} including the time integral on C[0,T].
url http://dx.doi.org/10.1155/2014/916423
work_keys_str_mv AT donghyuncho analoguesofconditionalwienerintegralswithdriftandinitialdistributiononafunctionspace
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