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Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed

Assyr Abdulle1, Grigorios A Pavliotis2, Andrea Zanoni1

  • 1ANMC, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.

Statistics and Computing
|May 9, 2022
PubMed
Summary
This summary is machine-generated.

We developed new methods for estimating drift in multiscale diffusion processes using discrete observations. Our techniques, based on homogenized dynamics, offer reliable and efficient drift estimation for complex systems.

Keywords:
Diffusion processDiscrete observationsEigenvalue problemFilteringHomogenizationLangevin dynamicsMartingale estimatorsParameter estimation

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Area of Science:

  • Stochastic processes
  • Mathematical physics
  • Computational mathematics

Background:

  • Accurate drift estimation is crucial for understanding and modeling complex dynamical systems.
  • Multiscale diffusion processes, particularly those governed by Langevin dynamics, present significant challenges in parameter estimation due to differing timescales.
  • Existing methods often struggle with the rate of data sampling, impacting the unbiasedness of drift estimators.

Purpose of the Study:

  • To propose novel methods for estimating the drift coefficient in multiscale diffusion processes from discrete observational data.
  • To leverage the spectral properties (eigenvalues and eigenfunctions) of homogenized dynamics for improved drift estimation.
  • To develop estimators that are robust to the sampling rate of observations.

Main Methods:

  • Utilizing eigenvalues and eigenfunctions of the homogenized dynamics associated with a two-scale potential.
  • Deriving a first drift estimator from a martingale estimating function of the homogenized diffusion process generator.
  • Introducing a second, asymptotically unbiased estimator that incorporates data filtering, independent of the sampling rate.

Main Results:

  • The first estimator's unbiasedness is dependent on the observation sampling rate.
  • The second estimator, incorporating data filtering, is proven to be asymptotically unbiased regardless of the sampling rate.
  • Numerical experiments demonstrate the reliability and efficiency of both proposed estimators.

Conclusions:

  • The developed methods provide effective tools for drift estimation in challenging multiscale diffusion scenarios.
  • The second estimator offers a significant advantage by being independent of the sampling rate, enhancing its practical applicability.
  • The findings contribute to the advancement of parameter estimation techniques for stochastic dynamical systems.