Deriving the Exact Discrete Analog of a Continuous Time System

Abstract

The exact discrete model satisfied by equispaced data generated by a linear stochastic differential equations system is derived by a method that does not imply restrictions on observed discrete data per se. The method involves integrating the solution of the continuous time model in state space form and a nonstandard change in the order of three types of integration, facilitating the representation of the exact discrete model as an asymptotically time-invariant vector autoregressive moving average model. The method applying to the state space form is general and is illustrated using the prototypical higher order model for mixed stock and flow data discussed by Bergstrom (1986, Econometric Theory 2, 350–373).

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

“…It is possible to evaluate these integrals involving the derivative Du(t) with the following integration-by-parts formula, used by Simos (1996) and established formally by McCrorie (2000).…”

Dynamics of the Almost Periodic Discrete Mackey–Glass Model

“…< − αe λ + e λ Q + βe λ Qe λτ = − αe λ + e λ + βe λ(τ+1) Q Hence, (16) and (20) imply that W( K) < Q. This contradicts that W( K) ≥ Q.…”

“…To the best of our observation, however, there are no published papers studying the discrete analogue of (2). There are many methods that can be used to derive the discrete equations from the continuous counterparts; see for instance the papers [19,20] for further details. By virtue of these methods, the discrete model of (2) can be viewed as:…”

Continuous Time Modelling Based on an Exact Discrete Time Representation

“…Owing to results by Van Loan (1978), the functions of the exponential can be computed as products of submatrices of a single, larger dimensional matrix exponential. Chambers (1999), McCrorie (2000) and Thornton and Chambers (2016) provide expressions pertaining to the exact discrete time model, while Harvey and Stock (1985) and Zadrozny (1988) provide similar expressions for application of the Kalman filter. Moler and Van Loan (1978), in a celebrated article in the numerical analysis literature, 22 showed that computation of the matrix exponential is a notoriously ill-conditioned problem, to the extent that of nineteen methods considered, only three or four were potentially suitable in general, including a scaling and squaring method that employs Padé approximation to the scalar exponential (see Higham, 2009).…”

“…th−h is now an MA(2) process which follows by noting that v th can be written under the white noise assumption as the sum of a pair of single intervals with respect to ζ(dr) over the intervals (th − 2h, th − h] and (th − h, th]; details can be found in McCrorie (2000). Although the autoregressive matrices remain the same functions of the underlying parameters as in the case of stock variables, the presence of flows affects the serial correlation properties of the disturbance vector, increasing the moving average order by one, a feature which needs to be incorporated in any estimation algorithm.…”