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Filtered data based estimators for stochastic processes driven by colored noise

Grigorios A. Pavliotis, Sebastian Reich, Andrea Zanoni

TL;DR

This paper studies inferring drift parameters in SDEs driven by colored noise when the correlation time tends to zero, i.e., the white-noise limit. It develops estimators based on filtered data for both maximum likelihood-type methods and stochastic gradient descent in continuous time (SGDCT), proving asymptotic unbiasedness and, in simple settings, asymptotic normality. The authors extend the framework to multiplicative colored noise where Lévy area corrections appear in the limit, establishing unbiasedness for the Lévy-corrected drift estimators. Numerical experiments corroborate the theory, showing that filtering is essential to remove bias and that both MLE and SGDCT can accurately recover drift coefficients, with MRLE often more robust in complex scenarios. Overall, the work provides a robust strategy for parameter inference from colored-noise data, with online learning capability and potential extensions to nonparametric drift and discrete observations.

Abstract

We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario.

Filtered data based estimators for stochastic processes driven by colored noise

TL;DR

This paper studies inferring drift parameters in SDEs driven by colored noise when the correlation time tends to zero, i.e., the white-noise limit. It develops estimators based on filtered data for both maximum likelihood-type methods and stochastic gradient descent in continuous time (SGDCT), proving asymptotic unbiasedness and, in simple settings, asymptotic normality. The authors extend the framework to multiplicative colored noise where Lévy area corrections appear in the limit, establishing unbiasedness for the Lévy-corrected drift estimators. Numerical experiments corroborate the theory, showing that filtering is essential to remove bias and that both MLE and SGDCT can accurately recover drift coefficients, with MRLE often more robust in complex scenarios. Overall, the work provides a robust strategy for parameter inference from colored-noise data, with online learning capability and potential extensions to nonparametric drift and discrete observations.

Abstract

We consider the problem of estimating unknown parameters in stochastic differential equations driven by colored noise, which we model as a sequence of Gaussian stationary processes with decreasing correlation time. We aim to infer parameters in the limit equation, driven by white noise, given observations of the colored noise dynamics. We consider both the maximum likelihood and the stochastic gradient descent in continuous time estimators, and we propose to modify them by including filtered data. We provide a convergence analysis for our estimators showing their asymptotic unbiasedness in a general setting and asymptotic normality under a simplified scenario.
Paper Structure (20 sections, 23 theorems, 205 equations, 10 figures)

This paper contains 20 sections, 23 theorems, 205 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Problem setting
  4. One-dimensional case
  5. Parameter estimation for additive colored noise
  6. Maximum likelihood type estimator
  7. The filtered data approach
  8. Coupling filtered data with SGDCT
  9. Convergence analysis
  10. Ergodic properties
  11. Infinite time limit of SGDCT
  12. Proof of the main results
  13. Lévy area correction
  14. Asymptotic unbiasedness for MLE estimator
  15. Asymptotic unbiasedness for SGDCT estimator
  16. ...and 5 more sections

Key Result

Theorem 3.4

Let as:ergodicityas:positive_definite$(i)$ hold and let $\widehat{\theta}(X,T)$ be defined in eq:MLE_ok. Then, it holds

Figures (10)

  • Figure 1: MLE estimator $\widehat{\theta}(X^\varepsilon,T)$ (left) and SGDCT estimator $\widetilde{\theta}^\varepsilon_t$ (right) in the one dimensional case with additive colored noise.
  • Figure 2: Four components of the MLE estimator $\widehat{\theta}(X,T)$ with data from the limit equation in presence of additive white noise.
  • Figure 3: Four components of the SGDCT estimator $\widetilde{\theta}_t$ with data from the limit equation in presence of additive white noise.
  • Figure 4: Four components of the MLE estimator $\widehat{\theta}_{\mathrm{exp}}^\delta(X,T)$ with filtered data from the limit equation in presence of additive white noise.
  • Figure 5: Four components of the SGDCT estimator $\widetilde{\theta}_{\mathrm{exp},t}^\delta$ with filtered data from the limit equation in presence of additive white noise.
  • ...and 5 more figures

Theorems & Definitions (57)

  • Example 2.1
  • Remark 2.2
  • Example 3.2
  • Remark 3.3
  • Theorem 3.4
  • Proposition 3.5
  • Remark 3.7
  • Remark 3.9
  • Remark 3.10
  • Theorem 3.11
  • ...and 47 more