4. The Concept of “Physical Realizability”#
We have asserted that the white noise “\(dN(t)/dt\)” does not exist as an ordinary stochastic process, it being so erratic that its variance is infinite. We have shown how this process can be regarded as a limiting point of a sequence of ordinary stochastic processes. It is useful to say a few words about the meaning of the concept of “physical realizability” of a stochastic process. We will use the concept of physical realizability to be interchangeable with the notion of an ordinary stochastic process.
We use the following definition:
Definition 7
A stochastic process is said to be ordinary or physically realizable if its “realizations” or “sample paths” can be represented as ordinary functions of time.
Loosely speaking, this means that it is in principle possible to “draw” each sample path as a function of time. (It may, however, sometimes take a long time to do so, since for example, the sample path of the Wiener process is not of bounded variation.)
We also use the following definition:
Definition 8
A generalized stochastic process is a stochastic process whose sample paths cannot be represented as ordinary functions, but only as limit points of sequences of ordinary functions. The sample paths of a “generalized stochastic process” can only be represented as “generalized functions.” The autocorrelation function of a generalized stochastic process will itself be a generalized function.
Loosely speaking, sample paths of a generalized stochastic process cannot be “drawn,” but can only be represented in terms of ideal “pulses” of zero width but positive “mass.” For this reason, a generalized stochastic process is said not to be “physically realizable.”
A generalized stochastic process is so erratic that it cannot be drawn. However, often there exists a moving average of a generalized stochastic process that is physically realizable. The shot-noise process \(Y(t)\) of 3. The Poisson Counting Process, formed by integrating the generalized white noise \(dN/dt\) against a square-integrable kernel, is the prototype: the noise lives only under the integral sign, while \(Y(t)\) itself is an ordinary process. In our economic models to be constructed below, generalized stochastic processes will appear only under integral signs. The economic models to be used always imply that the observable economic variables are physically realizable. This is exactly the role white noise plays in the moving-average (Wold) representation of 8. Spectral Densities, and the admissibility of a physically unrealizable white-noise input recurs in the estimation problem of 21. Inferring a Continuous-Time System from Discrete-Time Data: An Appreciation of A. W. Phillips (1959).