Globally optimal parameter estimates for nonlinear diffusions

TL;DR

This paper introduces an approximation method for the likelihood of nonlinear diffusion processes and an efficient EML algorithm that finds globally optimal parameters for a specific subclass of these processes.

Contribution

It develops a uniform convergence approximation for the likelihood function and proposes a novel EML algorithm for parameter inference in nonlinear SDEs with constant volatility.

Findings

The approximation converges uniformly to the true likelihood.

The EML algorithm efficiently finds globally optimal parameters.

Simulation results show competitive performance with existing methods.

Abstract

This paper studies an approximation method for the log-likelihood function of a nonlinear diffusion process using the bridge of the diffusion. The main result (Theorem \refthm:approx) shows that this approximation converges uniformly to the unknown likelihood function and can therefore be used efficiently with any algorithm for sampling from the law of the bridge. We also introduce an expected maximum likelihood (EML) algorithm for inferring the parameters of discretely observed diffusion processes. The approach is applicable to a subclass of nonlinear SDEs with constant volatility and drift that is linear in the model parameters. In this setting, globally optimal parameters are obtained in a single step by solving a linear system. Simulation studies to test the EML algorithm show that it performs well when compared with algorithms based on the exact maximum likelihood as well as closed-form likelihood expansions.