Red noise in continuous-time stochastic modelling.

Abstract

The concept of time-correlated noise is important to applied stochastic modelling. Nevertheless, there is no generally agreed-upon definition of the term red noise in continuous-time stochastic modelling settings. We present here a rigorous argumentation for the Ornstein-Uhlenbeck process integrated against time ( $U_{t}dt$ ) as a uniquely appropriate red noise implementation. We also identify the term $d{U}_t$ as an erroneous formulation of red noise commonly found in the applied literature. To this end, we prove a theorem linking properties of the power spectral density (PSD) to classes of Itô-differentials. The commonly ascribed red noise attribute of a PSD decaying as $S\left(\omega \right)\sim {\omega }^{-2}$ restricts the range of possible Itô-differentials $d{Y}_t={\alpha }_{t}dt+{\beta }_{t}d{W}_t$ . In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $d{Y}_t={\alpha }_{t}dt$ for almost all $t\ge 0$ . We further point out that taking $({\alpha }_t{)}_{t\ge 0}$ to be an Ornstein-Uhlenbeck process constitutes a uniquely relevant model choice due to its Gauss-Markov property. The erroneous use of the noise term $d{U}_t$ as red noise and its consequences are discussed in two examples from the literature.