It is shown that a mean zero Gaussian process has a representation in terms of stochastic integrals in the form
where the kernels are deterministic and the are independent Brownian motions or, more generally, Gaussian martingales with independent increments. Moreover, the are non- anticipating functions of X, i.e. is - measurable, where , . (This property is vital if one wants to use the representation in real time. The familiar spectral representation is not, as the author notes, non- anticipating.)
Set and define a Gaussian martingale by
The fundamental result is that for each t the , , , together with , . generate . The in (*) are then constructed from the 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 , , and the index of multiplicity, which is the dimension of the space of martingales spanned by , . Roughly speaking, the index of multiplicity gives the number of needed in (*), while the index of stationarity has to do with the structure of the kernels . If X is stationary, then , N(t) is constant, and for some function . Both indices are non- decreasing in t, E(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.