Least-action filtering

TL;DR

This paper introduces a variational approach to estimate hidden diffusion processes in continuous time by minimizing an action integral, providing a framework for understanding the distribution of the process around the estimated path.

Contribution

It develops a least-action filtering method using calculus of variations and asymptotic analysis, with numerical techniques for solving the resulting ODEs and handling local minima.

Findings

The least-action path approximates the true hidden process.

The actual path can be expressed as the sum of the least-action path and a Gaussian process.

Numerical methods are effective but must address local minima issues.

Abstract

This paper presents an approach to estimating a hidden process in a continuous-time setting, where the hidden process is a diffusion. The approach is simply to minimize the negative log-likelihood of the hidden path, where the likelihood is expressed relative to Wiener measure. This negative log-likelihood is the action integral of the path, which we minimize by calculus of variations. We then perform an asymptotic maximum-likelihood analysis to understand better how the actual path is distributed around the least-action path; it turns out that the actual path can be expressed (approximately) as the sum of the least-action path and a zero-mean Gaussian process which can be specified quite explicitly. Numerical solution of the ODEs which arise from the calculus of variations is often feasible, but is complicated by the shooting nature of the problem, and the possibility that we have found a local but not global minimum. We analyze the situations when this happens, and provide effective numerical methods for studying this. We also show how the methodology works in a situation where the hidden positive diffusion acts as the random intensity of a point process which is observed; here too it is possible to estimate the hidden process.