Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale. (19th April 2015)

• Record Type: Journal Article
• Authors: Wang, Caishi; Chen, Jinshu
• Other Names: Chung Jaeyoung Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2015(2015)
• Publication Date: 2015-04-19
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2015/714745
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : We aim at characterizing generalized functionals of discrete-time normal martingales. LetM = ( M n ) n ∈ N be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals ofM with an appropriate orthonormal basis forM '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.