A Discrete Approach to the Chaotic Representation Property

Abstract

In continuous time, let $(X_t)_{t{\geqslant}0}$ be a normal martingale (i.e. a process such that both X _t and X² _t -t are martingales). One says that X has the chaotic representation property if ${\rm L}^2(\sigma(X))$ is the (direct) Hilbert sum $\displaystyle\bigoplus_{p\in\mathbb{N}}\mathcal{X}_p(X),$ where $\mathcal{X}_p(X)$ is the space of all p-fold iterated stochastic integrals

$$\int_{0 < t_1 < \ldots < t_p} f(t_1,\ldots,t_p)dX_{t_1}\ldots dX_{t_p}$$

with f square-integrable ($\mathcal{X}_p(X)$ is called the p^th chaotic space; by convention $\mathcal{X}_0(X)$ is the one-dimensional space of deterministic random variables). An open problem is to characterize those processes X.

Instead of working in continuous time, we shall address an analogue of this problem where the time-axis is the set of $\mathbb{Z}$ of signed integers; in this setting, we shall give a sufficient (but probably far from necessary) condition for the chaotic representation property to hold.