Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift 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 Y:C[0,t]×[0,t]→ℝ by Y(x,s)=∫0sh(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t], where h∈L2[0,t] with h≠0 a.e. and a is continuous on [0,t]. Let random vectors Yn:...
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Wiley
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/469840 |
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| author | Dong Hyun Cho |
| author_facet | Dong Hyun Cho |
| author_sort | Dong Hyun Cho |
| collection | DOAJ |
| 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 Y:C[0,t]×[0,t]→ℝ by Y(x,s)=∫0sh(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t], where h∈L2[0,t] with h≠0 a.e. and a is continuous on [0,t]. Let random vectors Yn:C[0,t]→ℝn and Yn+1:C[0,t]→ℝn+1 be given by Yn(x)=(Y(x,t1),…,Y(x,tn)) and Yn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)), where 0<t1<⋯<tn<tn+1=t is a partition of [0,t]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove that Y is a generalized Brownian motion process with drift a. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,t] with the drift and the conditioning functions Yn and Yn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on C[0,t]. |
| format | Article |
| id | doaj-art-be4ed41d55304f0e878b10b9daea691d |
| institution | Kabale University |
| issn | 0972-6802 1758-4965 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces and Applications |
| spelling | doaj-art-be4ed41d55304f0e878b10b9daea691d2025-02-03T05:59:17ZengWileyJournal of Function Spaces and Applications0972-68021758-49652013-01-01201310.1155/2013/469840469840Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift 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 Y:C[0,t]×[0,t]→ℝ by Y(x,s)=∫0sh(u)dx(u)+a(s) for x∈C[0,t] and s∈[0,t], where h∈L2[0,t] with h≠0 a.e. and a is continuous on [0,t]. Let random vectors Yn:C[0,t]→ℝn and Yn+1:C[0,t]→ℝn+1 be given by Yn(x)=(Y(x,t1),…,Y(x,tn)) and Yn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)), where 0<t1<⋯<tn<tn+1=t is a partition of [0,t]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove that Y is a generalized Brownian motion process with drift a. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on C[0,t] with the drift and the conditioning functions Yn and Yn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on C[0,t].http://dx.doi.org/10.1155/2013/469840 |
| spellingShingle | Dong Hyun Cho Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space Journal of Function Spaces and Applications |
| title | Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space |
| title_full | Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space |
| title_fullStr | Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space |
| title_full_unstemmed | Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space |
| title_short | Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space |
| title_sort | evaluation formulas for generalized conditional wiener integrals with drift on a function space |
| url | http://dx.doi.org/10.1155/2013/469840 |
| work_keys_str_mv | AT donghyuncho evaluationformulasforgeneralizedconditionalwienerintegralswithdriftonafunctionspace |