Deriving the exact discrete analog of a continuous time system (Q2716479)

The paper presents a method of deriving an exact discrete model satisfied by equispaced data generated by an affine stochastic differential equation NEWLINE\[NEWLINEd\overline x(t)=\overline A(\theta)\overline x(t)dt+ d\overline \zeta (t),NEWLINE\]NEWLINE where \(\overline A(\theta)\in R^{n\times n}\), \(d\overline\zeta(t)\) is an \(n\)-dimensional zero-mean random measure with covariance function NEWLINE\[NEWLINEE\bigl[ \overline \zeta(\Delta_1) \otimes\overline \zeta(\Delta_2) \bigr]=|\Delta_1 \cap\Delta_2 |\Sigma (\mu),\quad \Delta_1, \Delta_2\in {\mathcal B}(R_+),\;\Sigma (\mu) \in R^{n \times n},NEWLINE\]NEWLINE and \(\theta,\mu\) are unknown structural parameters; stationarity or even stability of the solutions are not supposed. Using \textit{A.R. Bergström}'s existence and uniqueness result [Econometrica 51, 117-152 (1983, Zbl 0505.62071)], the method is based on an integration of \(\overline x\) and a change of the order of three types of integrals, and represents the exact discrete time model as an asymptotically time-invariant vector autoregressive moving average (VARMA) model. The author illustrates his method by its application to a prototypical higher order model for mixed stock and flow data discussed by Bergström.