Monte Carlo maximum likelihood estimation for discretely observed diffusion processes

TL;DR

This paper presents a Monte Carlo method for unbiased maximum likelihood estimation in discretely observed diffusion processes, achieving consistency and efficiency without discretization error, and providing guidelines for iteration tuning.

Contribution

It introduces an unbiased Monte Carlo MLE approach for diffusion models that converges almost surely and is computationally efficient, with theoretical and practical insights.

Findings

Monte Carlo MLE converges almost surely to the true parameter.

The method is unbiased and free of discretization error.

Optimal number of Monte Carlo iterations scales as √n.

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.