Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

TL;DR

This paper develops a framework for characterizing generalized functionals of discrete-time normal martingales using a new transform, the Fock transform, highlighting differences from continuous-time processes.

Contribution

It introduces the Fock transform for discrete-time normal martingales and establishes new characterization theorems based solely on growth conditions.

Findings

Characterized generalized functionals via the Fock transform.

Showed growth condition suffices for discrete-time martingales.

Contrasted discrete-time and continuous-time process characterizations.

Abstract

In this paper, we aim at characterizing generalized functionals of discrete-time normal martingales. Let $M=(M_n)_{n\in \mathbb{N}}$ 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.