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Zbl 0588.60029
Knight, F.B.
A post-predictive view of Gaussian processes. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 16, 541-566 (1983). ISSN 0012-9593

It is shown that a mean zero Gaussian process $\{X\sb t, -\infty <t<\infty \}$ has a representation in terms of stochastic integrals in the form $$ (*)\quad X\sb t=\sum\sb{n}\int\sp{t}\sb{0}F\sb n(t,s) dW\sp n\sb s $$ where the kernels $F\sb n$ are deterministic and the $W\sp n\sb t$ are independent Brownian motions or, more generally, Gaussian martingales with independent increments. Moreover, the $W\sp n$ are non- anticipating functions of X, i.e. $W\sp n\sb t$ is ${\cal F}\sb t$- measurable, where ${\cal F}\sb t=\sigma \{X\sb s$, $s\le t\}$. (This property is vital if one wants to use the representation in real time. The familiar spectral representation $X\sb t=\int\sp{\infty}\sb{- \infty}c(u) c\sp{2\pi itu} dZ\sb u$ is not, as the author notes, non- anticipating.) \par Set $P\sb{\lambda}(t)=\lambda E\{\int\sp{\infty}\sb{0}e\sp{-\lambda s} X\sb{t+s} ds\vert {\cal F}\sb{t+}\}$ and define a Gaussian martingale $M\sb{\lambda}$ by $$ M\sb{\lambda}(t)=P\sb{\lambda}(t)- P\sb{\lambda}(0)+\int\sp{t}\sb{0}(X\sb u-P\sb{\lambda}(u))du,\quad t\ge 0. $$ The fundamental result is that for each t the $M\sb{\lambda}(s)$, $s\le t$, $\lambda =1,2,3,...$, together with $P\sb{\lambda}(0)$, $\lambda =1,2,3,..$. generate ${\cal F}\sb t$. The $W\sp n$ in (*) are then constructed from the $M\sb{\lambda}$ by a sort of Gram-Schmidt procedure. \par The analysis involves a careful examination of two indices: the index of stationarity N(t), which is the dimension of the linear space spanned by the random variables $\{M\sb{\lambda}(t)$, $\lambda >0\}$, and the index of multiplicity, which is the dimension of the space of martingales spanned by $\{M\sb{\lambda}(s)$, $s\le t\}$. Roughly speaking, the index of multiplicity gives the number of $W\sp n$ needed in (*), while the index of stationarity has to do with the structure of the kernels $F\sb n(t,s)$. If X is stationary, then $E(t)=1$, N(t) is constant, and $F\sb n(t,s)=f\sb n(t-s)$ for some function $f\sb n$. Both indices are non- decreasing in t, E(t)$\le N(t)$, and either or both may be infinite. \par A number of examples are worked out to illustrate the possibilities and to show how to find the representation in various specific cases.
[J.Walsh]

MSC 2000:: *60G15 Gaussian processes
60G25 Prediction theory
60H99 Stochastic analysis

Keywords: Gaussian martingales with independent increments; spectral representation; index of stationarity; index of multiplicity

Cited in: Zbl 0588.60030

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Zentralblatt MATH Berlin [Germany]