Red noise in continuous-time stochastic modelling
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916894083121152 |
|---|---|
| author | Morr, Andreas Kreher, Dörte Boers, Niklas |
| author_facet | Morr, Andreas Kreher, Dörte Boers, Niklas |
| contents | 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 Itô-differentials. The commonly ascribed red noise attribute of a PSD decaying as $S(ω)\simω^{-2}$ restricts the range of possible Itô-differentials $\mathrm{d}Y_t=α_t\mathrm{d} t+β_t\mathrm{d} W_t$. In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d}Y_t=α_t\mathrm{d} t$ for almost all $t\geq 0$. We further point out that taking $(α_t)_{t\geq 0}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03566 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Red noise in continuous-time stochastic modelling Morr, Andreas Kreher, Dörte Boers, Niklas Probability Dynamical Systems 37H05, 37A50, 60H10, 60G10, 60G15 G.3 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 Itô-differentials. The commonly ascribed red noise attribute of a PSD decaying as $S(ω)\simω^{-2}$ restricts the range of possible Itô-differentials $\mathrm{d}Y_t=α_t\mathrm{d} t+β_t\mathrm{d} W_t$. In particular, any such differential with continuous, square-integrable integrands must have a vanishing martingale part, i.e. $\mathrm{d}Y_t=α_t\mathrm{d} t$ for almost all $t\geq 0$. We further point out that taking $(α_t)_{t\geq 0}$ 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. |
| title | Red noise in continuous-time stochastic modelling |
| topic | Probability Dynamical Systems 37H05, 37A50, 60H10, 60G10, 60G15 G.3 |
| url | https://arxiv.org/abs/2212.03566 |