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Zbl 0982.60031
Émery, M.
A discrete approach to the chaotic representation property. (English)
[A] Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXXV. Berlin: Springer. Lect. Notes Math. 1755, 123-138 (2001). ISBN 3-540-41659-5

Author's abstract: In continuous time, let $(X_t)_{t\ge 0}$ 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 $\bigoplus_{p\in {\Bbb N}} \chi_p(X)$, 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 $\Bbb Z$ of signed integers; in this setting, we shall give a sufficient (but probably far from necessary) condition for the chaotic representation property to hold.
[Tomasz Bojdecki (Warszawa)]

MSC 2000:: *60G42 Martingales with discrete parameter

Keywords: normalized martingale increment (novation process); chaotic representation property

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