Red Noise in continuous-time Stochastic ModellingMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

The concept of time-correlated noise is an important tool of both discrete- and continuous-time stochastic modelling but approaches to its implementation are manifold. More specifically, while the discrete-time notion of so-called red noise is often synonymous with AR(1)-noise, there has not yet been a comprehensive motivation for the use of any specific continuous-time analogue. We discuss this generalisation to the continuous-time case and its inherent ambiguities. The implications of carrying certain attributes like the autocovariance structure from the discrete-time to the continuous-time setting are explored. We then exploit the characterisation of red noise via its power spectral density to narrow down the range of feasible model choices. We find that the attribute of a power spectral density decaying as $S(\omega)\sim\omega^{-2}$ commonly ascribed to the notion of red noise has far reaching consequences when posited in the continuous-time stochastic differential setting. In particular, any such It\^{o}-differential $\mathrm{d} Y_t=\alpha_t \mathrm{d} t+\beta_t \mathrm{d} W_t$ 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 argue that $\alpha$ should be an Ornstein-Uhlenbeck process.