Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale
We aim at characterizing generalized functionals of discrete-time normal martingales. Let M=(Mn)n∈N be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of M with an appropriate orthonormal basis for M’s square inte...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Journal of Function Spaces |
| Online Access: | http://dx.doi.org/10.1155/2015/714745 |
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| author | Caishi Wang Jinshu Chen |
| author_facet | Caishi Wang Jinshu Chen |
| author_sort | Caishi Wang |
| collection | DOAJ |
| description | We aim at characterizing generalized functionals of discrete-time normal martingales. Let M=(Mn)n∈N be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of M with an appropriate orthonormal basis for M’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes. |
| format | Article |
| id | doaj-art-593f0550560b4ed88b1cf8a073f6cdeb |
| institution | Kabale University |
| issn | 2314-8896 2314-8888 |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Function Spaces |
| spelling | doaj-art-593f0550560b4ed88b1cf8a073f6cdeb2025-02-03T01:10:52ZengWileyJournal of Function Spaces2314-88962314-88882015-01-01201510.1155/2015/714745714745Characterization Theorems for Generalized Functionals of Discrete-Time Normal MartingaleCaishi Wang0Jinshu Chen1School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, ChinaSchool of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, ChinaWe aim at characterizing generalized functionals of discrete-time normal martingales. Let M=(Mn)n∈N be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of M with an appropriate orthonormal basis for M’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes.http://dx.doi.org/10.1155/2015/714745 |
| spellingShingle | Caishi Wang Jinshu Chen Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale Journal of Function Spaces |
| title | Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale |
| title_full | Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale |
| title_fullStr | Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale |
| title_full_unstemmed | Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale |
| title_short | Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale |
| title_sort | characterization theorems for generalized functionals of discrete time normal martingale |
| url | http://dx.doi.org/10.1155/2015/714745 |
| work_keys_str_mv | AT caishiwang characterizationtheoremsforgeneralizedfunctionalsofdiscretetimenormalmartingale AT jinshuchen characterizationtheoremsforgeneralizedfunctionalsofdiscretetimenormalmartingale |