Red noise in continuous-time stochastic modelling

TL;DR

This paper rigorously defines red noise in continuous-time stochastic models, advocating for the Ornstein-Uhlenbeck process integrated against time as the proper implementation, and clarifies misconceptions about the use of differential terms.

Contribution

It provides a rigorous theoretical framework linking spectral properties to Itô-differentials, and clarifies the correct modeling of red noise in continuous-time stochastic processes.

Findings

Ornstein-Uhlenbeck process integrated against time is the appropriate red noise model.

Differential $ ext{d}U_t$ is an erroneous formulation of red noise.

Red noise with PSD decaying as $requency^{-2}$ restricts the form of Itô-differentials.

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.