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 It\^{o}-differentials. The commonly ascribed red noise attribute of a PSD decaying as $S(\omega)\sim\omega^{-2}$ restricts the range of possible It\^{o}-differentials $\mathrm{d}Y_t=\alpha_t\mathrm{d} t+\beta_t\mathrm{d} W_t$. In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d}Y_t=\alpha_t\mathrm{d} t$ for almost all $t\geq 0$. We further point out that taking $(\alpha_t)_{t\geq 0}$ 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.