Continuous-time autoregressive moving average processes in discrete time: representation and embeddability


Correspondence to: Michael A. Thornton, Department of Economics and Related Studies, University of York, Heslington, YO10 5DD, UK.


This article explores techniques to derive the exact discrete-time representation for data generated by a continuous-time autoregressive moving average (ARMA) process, augmenting existing methods with a stochastic integration-by-parts formula. The continuous-time ARMA(2, 1) system is considered in detail, and a mapping from the parameters of a univariate discrete-time ARMA(2, 1) process to a univariate continuous-time ARMA(2, 1) process observed at discrete intervals is derived. This is used to derive conditions for the embeddability of such processes.