Online Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes
This work addresses the challenge of designing adaptive filters and optimal controllers for unknown or changing systems in fields like robotics, neuroscience, and finance, providing theoretical convergence analysis where it was previously lacking.
The authors tackled the problem of online parameter estimation for partially observed diffusion processes by analyzing a stochastic gradient ascent algorithm on the incomplete-data log-likelihood, proving its convergence under ergodicity conditions and demonstrating numerically that it can improve suboptimal filters and handle non-identifiable systems.
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.