*(English)*Zbl 0588.60029

It is shown that a mean zero Gaussian process $\{{X}_{t},-\infty <t<\infty \}$ has a representation in terms of stochastic integrals in the form

where the kernels ${F}_{n}$ are deterministic and the ${W}_{t}^{n}$ are independent Brownian motions or, more generally, Gaussian martingales with independent increments. Moreover, the ${W}^{n}$ are non- anticipating functions of X, i.e. ${W}_{t}^{n}$ is ${\mathcal{F}}_{t}$- measurable, where ${\mathcal{F}}_{t}=\sigma \{{X}_{s}$, $s\le t\}$. (This property is vital if one wants to use the representation in real time. The familiar spectral representation ${X}_{t}={\int}_{-\infty}^{\infty}c\left(u\right){c}^{2\pi itu}d{Z}_{u}$ is not, as the author notes, non- anticipating.)

Set ${P}_{\lambda}\left(t\right)=\lambda E\left\{{\int}_{0}^{\infty}{e}^{-\lambda s}{X}_{t+s}ds\right|{\mathcal{F}}_{t+}\}$ and define a Gaussian martingale ${M}_{\lambda}$ by

The fundamental result is that for each t the ${M}_{\lambda}\left(s\right)$, $s\le t$, $\lambda =1,2,3,\xb7\xb7\xb7$, together with ${P}_{\lambda}\left(0\right)$, $\lambda =1,2,3,\xb7\xb7$. generate ${\mathcal{F}}_{t}$. The ${W}^{n}$ in (*) are then constructed from the ${M}_{\lambda}$ by a sort of Gram-Schmidt procedure.

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 $\{{M}_{\lambda}\left(t\right)$, $\lambda >0\}$, and the index of multiplicity, which is the dimension of the space of martingales spanned by $\{{M}_{\lambda}\left(s\right)$, $s\le t\}$. Roughly speaking, the index of multiplicity gives the number of ${W}^{n}$ needed in (*), while the index of stationarity has to do with the structure of the kernels ${F}_{n}(t,s)$. If X is stationary, then $E\left(t\right)=1$, N(t) is constant, and ${F}_{n}(t,s)={f}_{n}(t-s)$ for some function ${f}_{n}$. Both indices are non- decreasing in t, E(t)$\le N\left(t\right)$, and either or both may be infinite.

A number of examples are worked out to illustrate the possibilities and to show how to find the representation in various specific cases.