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 pth 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.