Parameter estimation for partially observed second-order diffusion processes

TL;DR

This paper introduces a simple modification to stochastic gradient descent and Kalman filter methods for parameter estimation in partially observed second-order diffusion processes, effectively removing systematic biases without complex corrections.

Contribution

It proposes a straightforward modification to existing estimation algorithms that eliminates biases in partially observed second-order diffusion processes.

Findings

The modified methods remove systematic estimation biases.

The approach extends to maximum likelihood estimation.

It simplifies existing correction procedures.

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.