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A post-predictive view of Gaussian processes. (English) Zbl 0588.60029

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

(*)X t = n 0 t F n (t,s)dW s n

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 t - measurable, where t =σ{X s , st}. (This property is vital if one wants to use the representation in real time. The familiar spectral representation X t = - c(u)c 2πitu dZ u is not, as the author notes, non- anticipating.)

Set P λ (t)=λE{ 0 e -λs X t+s ds| t+ } and define a Gaussian martingale M λ by

M λ (t)=P λ (t)-P λ (0)+ 0 t (X u -P λ (u))du,t0·

The fundamental result is that for each t the M λ (s), st, λ=1,2,3,···, together with P λ (0), λ=1,2,3,··. generate t . The W n in (*) are then constructed from the M λ 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 λ (t), λ>0}, and the index of multiplicity, which is the dimension of the space of martingales spanned by {M λ (s), st}. 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(t)=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)N(t), 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.

Reviewer: J.Walsh

MSC:
60G15Gaussian processes
60G25Prediction theory
60H99Stochastic analysis