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A Discrete Approach to the Chaotic Representation Property

Part of the Lecture Notes in Mathematics book series (SEMPROBAB,volume 1755)

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.

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.

Mathematics Subject Classification (2000):

  • 60GXX
  • 60HXX
  • 60JXX

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Correspondence to M. Émery .

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© 2001 Springer-Verlag Berlin/Heidelberg

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Émery, M. (2001). A Discrete Approach to the Chaotic Representation Property. In: Azéma, J., Émery, M., Ledoux, M., Yor, M. (eds) Séminaire de Probabilités XXXV. Lecture Notes in Mathematics, vol 1755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-44671-2_7

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  • DOI: https://doi.org/10.1007/978-3-540-44671-2_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41659-3

  • Online ISBN: 978-3-540-44671-2

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