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  3. Red Noise In Continuous-time Stochastic Modelling.
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  3. Red Noise In Continuous-time Stochastic Modelling.

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Red noise in continuous-time stochastic modelling.

Andreas Morr1,2, Dörte Kreher3, Niklas Boers1,2,4

  • 1Department of Aerospace and Geodesy, TUM School of Engineering and Design, Munich, Bavaria, Germany.

Royal Society Open Science
|August 14, 2025

View abstract on PubMed

Summary
This summary is machine-generated.

This study rigorously defines red noise in continuous-time stochastic modeling, proposing the integrated Ornstein-Uhlenbeck process as the correct implementation. It corrects the common misuse of "dU_t" as red noise, crucial for accurate time-correlated noise modeling.

Keywords:
continuous-time modellingcorrelated noisered noisestochastic modelling

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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.