The Correspondence Between Stochastic Linear Difference and Differential Equations

The relationship between autoregressive moving-average (ARMA) models in discrete time and the corresponding models in continuous time is examined in this paper. The linear stochastic models that are commonly regarded as the counterparts of the ARMA models are driven by a forcing function that consists of the increments of a Wiener Process. This function is unbounded in frequency. In cases where the periodogram of the data indicates that there is a clear upper bound to its frequency content, we propose an alternative frequency- limited white-noise forcing function. Then, there is a straightforward translation from the ARMA model to a differential equation, which is based of the principle of impulse invariance. Whenever there is no perceptible limit to the frequency content, the translation must be based on a principle of autocovariance equivalence. On the website of the author, there is a computer program that effects both of these discrete-to-continuous translations.