Title: Stochastic processes in discrete and continuous time
Authors: Stephen Pollock - University of Leicester (United Kingdom) [presenting]
Abstract: The relationship is considered between discrete-time ARMA models and the corresponding stochastic differential equations in continuous time. The precise relationship depends on the assumptions that are made regarding the forcing function. In the classical theory of stochastic differential equations, the forcing function is a continuous-time white-noise process, which is derived from the increments of a Weiner process that is unbounded in frequency. We develop a theory that accommodates forcing functions that are limited in frequency. The forcing function of a discrete-time ARMA model is limited in frequency to the Nyquist interval $[-p_i, p_i]$, where the frequency is measured in radians per unit interval. A one-to-one correspondence with a continuous model can be established. A continuous trajectory is created by replacing the sequence discrete-time ordinates by a superposition of suitably scaled sinc functions at unit displacements. The stochastic differential equation for such a frequency-limited continuous-time process can be derived readily. An application to business-cycle analysis is presented.