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 ($U_t\mathrm{d}t$) as a uniquely appropriate red noise implementation. We also identify the term $\mathrm{d}{U}_t$ 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 $S(\ddot{I}‰)\sim \ddot{I}{‰}^{-2}$ restricts the range of possible Itô-differentials $\mathrm{d}{Y}_t=\hat{I}{\pm }_t\mathrm{d}t+{\hat{I}}_t^2\mathrm{d}{W}_t$. In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d}{Y}_t=\hat{I}{\pm }_t\mathrm{d}t$ for almost all $t\ge 0$. We further point out that taking $(\hat{I}{\pm }_t{)}_{t\ge 0}$ 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 $\mathrm{d}{U}_t$ as red noise and its consequences are discussed in two examples from the literature.