Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Parameter estimation for partially observed second-order diffusion processes

Jan Albrecht, Sebastian Reich

TL;DR

This work addresses online parameter estimation for second-order diffusion processes when only position observations are available, a setting in which traditional SGD and Kalman-filter approaches become biased due to temporal correlations in the driving noise. The authors introduce a simple, principled forward-shifted innovation modification that preserves an Itô-type martingale structure, yielding unbiased online SGD and Kalman-filter updates; they illustrate the approach with a motivating OU-based example and a nonlinear test case, showing convergence to true parameters where standard methods fail. The paper also discusses a corrected MLE formulation and highlights potential extensions to multi-scale diffusion settings and noisy observations in the position measurements. Overall, the proposed corrections improve robustness of online parameter learning for partially observed second-order diffusions and offer practical tools for real-time inference in noisy, temporally correlated systems.

Abstract

Estimating parameters of a diffusion process given continuous-time observations of the process via maximum likelihood approaches or, online, via stochastic gradient descent or Kalman filter formulations constitutes a well-established research area. It has also been established previously that these techniques are, in general, not robust to perturbations in the data in the form of temporal correlations of the driving noise. While the subject is relatively well understood and appropriate modifications have been suggested in the context of multi-scale diffusion processes and their reduced model equations, we consider here an alternative but related setting where a diffusion process in positions and velocities is only observed via its positions. In this note, we propose a simple modification to standard stochastic gradient descent and Kalman filter formulations, which eliminates the arising systematic estimation biases. The modification can be extended to standard maximum likelihood approaches and avoids computation of previously proposed correction terms.

Parameter estimation for partially observed second-order diffusion processes

TL;DR

This work addresses online parameter estimation for second-order diffusion processes when only position observations are available, a setting in which traditional SGD and Kalman-filter approaches become biased due to temporal correlations in the driving noise. The authors introduce a simple, principled forward-shifted innovation modification that preserves an Itô-type martingale structure, yielding unbiased online SGD and Kalman-filter updates; they illustrate the approach with a motivating OU-based example and a nonlinear test case, showing convergence to true parameters where standard methods fail. The paper also discusses a corrected MLE formulation and highlights potential extensions to multi-scale diffusion settings and noisy observations in the position measurements. Overall, the proposed corrections improve robustness of online parameter learning for partially observed second-order diffusions and offer practical tools for real-time inference in noisy, temporally correlated systems.

Abstract

Estimating parameters of a diffusion process given continuous-time observations of the process via maximum likelihood approaches or, online, via stochastic gradient descent or Kalman filter formulations constitutes a well-established research area. It has also been established previously that these techniques are, in general, not robust to perturbations in the data in the form of temporal correlations of the driving noise. While the subject is relatively well understood and appropriate modifications have been suggested in the context of multi-scale diffusion processes and their reduced model equations, we consider here an alternative but related setting where a diffusion process in positions and velocities is only observed via its positions. In this note, we propose a simple modification to standard stochastic gradient descent and Kalman filter formulations, which eliminates the arising systematic estimation biases. The modification can be extended to standard maximum likelihood approaches and avoids computation of previously proposed correction terms.
Paper Structure (7 sections, 39 equations, 1 figure)

This paper contains 7 sections, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivating example
  3. Unbiased SGD
  4. Unbiased Kalman filter
  5. Numerical example
  6. Conclusions
  7. Acknowledgments.

Figures (1)

  • Figure 1: Left panel: Estimated parameter value from standard SGD as a function of the iteration index. The iteration starts at $\theta_0 = 2$ and the reference value is $\theta_\ast = 1$. A systematic bias can clearly be seen. Middle panel: Same data is used in the modified SGD. The modified SGD algorithm converges to the correct reference value. Right panel: Same data is used in the unbiased Kalman filter with prior distribution $N(2, 1)$ and $\sigma$ treated as part of the prior. The lightly colored lines indicate the $1\sigma$-intervals of the Bayesian estimates.