Online Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes

TL;DR

This paper analyzes the convergence of an online stochastic gradient algorithm for estimating parameters in partially observed diffusion processes, providing theoretical guarantees and demonstrating practical improvements in filtering applications.

Contribution

It offers a theoretical analysis of the convergence of online maximum likelihood estimation for partially observed diffusions, including conditions and numerical validation.

Findings

Proves convergence of the stochastic gradient algorithm under ergodicity conditions

Shows potential for improving suboptimal filters in practice

Applicable even with non-identifiable systems

Abstract

We revisit the problem of estimating the parameters of a partially observed diffusion process, consisting of a hidden state process and an observed process, with a continuous time parameter. The estimation is to be done online, i.e. the parameter estimate should be updated recursively based on the observation filtration. We provide a theoretical analysis of the stochastic gradient ascent algorithm on the incomplete-data log-likelihood. The convergence of the algorithm is proved under suitable conditions regarding the ergodicity of the process consisting of state, filter, and tangent filter. Additionally, our parameter estimation is shown numerically to have the potential of improving suboptimal filters, and can be applied even when the system is not identifiable due to parameter redundancies. Online parameter estimation is a challenging problem that is ubiquitous in fields such as robotics, neuroscience, or finance in order to design adaptive filters and optimal controllers for unknown or changing systems. Despite this, theoretical analysis of convergence is currently lacking for most of these algorithms. This article sheds new light on the theory of convergence in continuous time.