Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises
The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let Z be a discrete-time normal noise that h...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2018/2954695 |
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| _version_ | 1849691723783995392 |
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| author | Caishi Wang Shuai Lin Ailing Huang |
| author_facet | Caishi Wang Shuai Lin Ailing Huang |
| author_sort | Caishi Wang |
| collection | DOAJ |
| description | The Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let Z be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of Z. Then, we use the Fock transform to define some fundamental operators on generalized functionals of Z and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of Z and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of Z. |
| format | Article |
| id | doaj-art-e17374c199e4490a96a5c983684a496d |
| institution | DOAJ |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-e17374c199e4490a96a5c983684a496d2025-08-20T03:20:58ZengWileyJournal of Function Spaces2314-88962314-88882018-01-01201810.1155/2018/29546952954695Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal NoisesCaishi Wang0Shuai Lin1Ailing Huang2School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, ChinaSchool of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, ChinaSchool of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, ChinaThe Clark-Ocone formula in the theory of discrete-time chaotic calculus holds only for square integrable functionals of discrete-time normal noises. In this paper, we aim at extending this formula to generalized functionals of discrete-time normal noises. Let Z be a discrete-time normal noise that has the chaotic representation property. We first prove a result concerning the regularity of generalized functionals of Z. Then, we use the Fock transform to define some fundamental operators on generalized functionals of Z and apply the abovementioned regularity result to prove the continuity of these operators. Finally, we establish the Clark-Ocone formula for generalized functionals of Z and show its application results, which include the covariant identity result and the variant upper bound result for generalized functionals of Z.http://dx.doi.org/10.1155/2018/2954695 |
| spellingShingle | Caishi Wang Shuai Lin Ailing Huang Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises Journal of Function Spaces |
| title | Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises |
| title_full | Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises |
| title_fullStr | Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises |
| title_full_unstemmed | Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises |
| title_short | Clark-Ocone Formula for Generalized Functionals of Discrete-Time Normal Noises |
| title_sort | clark ocone formula for generalized functionals of discrete time normal noises |
| url | http://dx.doi.org/10.1155/2018/2954695 |
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