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...

Full description

Saved in:
Bibliographic Details
Main Authors: Caishi Wang, Jinshu Chen
Format: Article
Language:English
Published: Wiley 2015-01-01
Series:Journal of Function Spaces
Online Access:http://dx.doi.org/10.1155/2015/714745
Tags: Add Tag
No Tags, Be the first to tag this record!
_version_ 1832564472013651968
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