A Discrete Approach to the Chaotic Representation Property
A Discrete Approach to the Chaotic Representation Property
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DOI：
10.1007/978-3-540-44671-2_7
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发表时间：
2001
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影响因子：
--
通讯作者：
M. Émery
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文献类型：
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作者：
M. Émery
In continuous time, let $ ( X t ) t⩾0 $$(X_t)_{t{\geqslant}0}$ be a normal martingale (i.e. a process such that bothXtandX2t-tare martingales). One says thatXhas thechaotic representation propertyif $ L 2 (σ(X)) $${\rm L}^2(\sigma(X))$ is the (direct) Hilbert sum $ ⊕ p∈ℕ X p (X), $$\displaystyle\bigoplus_{p\in\mathbb{N}}\mathcal{X}_p(X),$ where $ X p (X) $$\mathcal{X}_p(X)$ is the space of allp-fold iterated stochastic integrals $ ∫ 0< t 1 <…< t p f ( t 1 ,…, t p )d X t 1 …d X t p $$$\int_{0 < t_1 < \ldots < t_p} f(t_1,\ldots,t_p)dX_{t_1}\ldots dX_{t_p}$$ withfsquare-integrable ($ X p (X) $$\mathcal{X}_p(X)$ is called thepthchaotic space; by convention $ X 0 (X) $$\mathcal{X}_0(X)$ is the one-dimensional space of deterministic random variables). An open problem is to characterize those processesX.Instead of working in continuous time, we shall address an analogue of this problem where the time-axis is the set of $ ℤ $$\mathbb{Z}$ of signed integers; in this setting, we shall give a sufficient (but probably far from necessary) condition for the chaotic representation property to hold.
In continuous time, let $ ( X t ) t⩾0 $$(X_t)_{t{\geqslant}0}$ be a normal martingale (i.e. a process such that bothXtandX2t-tare martingales). One says thatXhas thechaotic representation propertyif $ L 2 (σ(X)) $${\rm L}^2(\sigma(X))$ is the (direct) Hilbert sum $ ⊕ p∈ℕ X p (X), $$\displaystyle\bigoplus_{p\in\mathbb{N}}\mathcal{X}_p(X),$ where $ X p (X) $$\mathcal{X}_p(X)$ is the space of allp-fold iterated stochastic integrals $ ∫ 0< t 1 <…< t p f ( t 1 ,…, t p )d X t 1 …d X t p $$$\int_{0 < t_1 < \ldots < t_p} f(t_1,\ldots,t_p)dX_{t_1}\ldots dX_{t_p}$$ withfsquare-integrable ($ X p (X) $$\mathcal{X}_p(X)$ is called thepthchaotic space; by convention $ X 0 (X) $$\mathcal{X}_0(X)$ is the one-dimensional space of deterministic random variables). An open problem is to characterize those processesX.Instead of working in continuous time, we shall address an analogue of this problem where the time-axis is the set of $ ℤ $$\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.