A discrete approach to the chaotic representation property. (English) Zbl 0982.60031

Émery, M.

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXXV. Berlin: Springer. Lect. Notes Math. 1755, 123-138 (2001).

Author’s abstract: In continuous time, let \((X_t)_{t\geq 0}\) be a normal martingale (i.e. a process such that both \(X_t\) and \(X_t^2-t\) are martingales). One says that \(X\) has the chaotic representation property if \(L^2(\sigma(X))\) is the direct Hilbert sum \(\bigoplus_{p\in {\mathbb N}} \chi_p(X)\), where \(\chi_p(X)\) is the space of all \(p\)-fold iterated stochastic integrals \(\int_{0<t_1<\dots<t_p}f(t_1,\dots,t_p) dX_{t_1}\dots dX_{t_p}\) with \(f\) square-integrable. 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 \(\mathbb Z\) of signed integers; in this setting, we shall give a sufficient (but probably far from necessary) condition for the chaotic representation property to hold.
For the entire collection see [Zbl 0960.00020].

Reviewer: Tomasz Bojdecki (Warszawa)

MSC:

60G42

Martingales with discrete parameter

Keywords:

normalized martingale increment (novation process); chaotic representation property