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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fthindawi:oai:hindawi.com:10.1155/2014/916423 2023-05-15T16:01:58+02:00 Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space Dong Hyun Cho 2014 https://doi.org/10.1155/2014/916423 en eng Abstract and Applied Analysis https://doi.org/10.1155/2014/916423 Copyright © 2014 Dong Hyun Cho. Research Article 2014 fthindawi https://doi.org/10.1155/2014/916423 2019-05-25T23:39:52Z 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]. Article in Journal/Newspaper DML Hindawi Publishing Corporation Abstract and Applied Analysis 2014 1 12 |
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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]. |
format |
Article in Journal/Newspaper |
author |
Dong Hyun Cho |
spellingShingle |
Dong Hyun Cho Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space |
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 |
Abstract and Applied Analysis |
publishDate |
2014 |
url |
https://doi.org/10.1155/2014/916423 |
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DML |
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DML |
op_relation |
https://doi.org/10.1155/2014/916423 |
op_rights |
Copyright © 2014 Dong Hyun Cho. |
op_doi |
https://doi.org/10.1155/2014/916423 |
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Abstract and Applied Analysis |
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2014 |
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1 |
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12 |
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1766397630854725632 |