Monte Carlo maximum likelihood estimation for discretely observed diffusion processes

Abstract

This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s.\@ continuous estimators of the likelihood function for a family of diffusion models and its performance in numerical examples is computationally efficient. It uses a recently developed technique for the exact simulation of diffusions, and involves no discretization error. We show that, under regularity conditions, the Monte Carlo MLE converges a.s. to the true MLE. For datasize $n\to\infty$, we show that the number of Monte Carlo iterations should be tuned as $\mathcal{O}(n^{1/2})$ and we demonstrate the consistency properties of the Monte Carlo MLE as an estimator of the true parameter value.