Red noise in continuous-time stochastic modelling

Area of Science:

• Stochastic modeling
• Time-correlated noise analysis
• Continuous-time processes

Background:

• The term "red noise" lacks a standardized definition in continuous-time stochastic modeling.
• Time-correlated noise is a critical concept in various applied fields.
• Existing literature often misuses specific formulations for red noise.

Purpose of the Study:

• To rigorously define and identify an appropriate implementation of red noise in continuous-time stochastic modeling.
• To correct the erroneous use of the Ornstein-Uhlenbeck process differential (dU_t) as red noise.
• To establish a theoretical link between power spectral density properties and Itô-differentials.

Main Methods:

• Mathematical proof linking power spectral density (PSD) properties to Itô-differentials.
• Analysis of Itô-differentials with a PSD decaying as S(ω) ~ ω^-2.
• Demonstration of the vanishing martingale part for specific Itô-differentials.
• Identification of the integrated Ornstein-Uhlenbeck process as a suitable red noise model.

Main Results:

• The integrated Ornstein-Uhlenbeck process (∫U_t dt) is rigorously established as the correct red noise implementation.
• The formulation dU_t is identified as an erroneous representation of red noise.
• Itô-differentials exhibiting a red noise PSD must have a vanishing martingale part.
• The Ornstein-Uhlenbeck process itself is highlighted for its Gauss-Markov property, making it a relevant choice.

Conclusions:

• The integrated Ornstein-Uhlenbeck process provides a uniquely appropriate definition for red noise in continuous-time stochastic modeling.
• Misapplication of dU_t as red noise can lead to inaccuracies in applied stochastic models.
• Understanding the relationship between PSD and Itô-differentials is key to correct noise modeling.