Globally Optimal Parameter Estimates for Non-Linear Diffusions

Abstract

This paper studies an approximation method for the log likelihood function of a non-linear diffusion process using the bridge of the diffusion. The main result (Theorem 1) 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 non-linear 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 square linear system whose dimension equals the number of parameters in the model. 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.

Keywords: Maximum likelihood, global optimization, non-linear diffusion, EM algorithm, estimation

JEL Classification: C13, C15