  The concept of correlated noise is well-established in discrete-time
stochastic modelling but there is no generally agreed-upon definition of the
notion of red noise in continuous-time stochastic modelling. Here we discuss
the generalization of discrete-time correlated noise to the continuous case. We
give an overview of existing continuous-time approaches to model red noise,
which relate to their discrete-time analogue via characteristics like the
autocovariance structure or the power spectral density. The implications of
carrying certain attributes from the discrete-time to the continuous-time
setting are explored while assessing the inherent ambiguities in such a
generalization. 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 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.

Probability;Dynamical Systems