Unbiased Estimation of the Hessian for Partially Observed Diffusions

TL;DR

This paper introduces an unbiased estimator for the Hessian of the log-likelihood in partially observed diffusion processes, overcoming bias from discretization and verified through numerical experiments on various models.

Contribution

It develops a novel unbiased Hessian estimator using Girsanov's theorem and randomization, applicable to complex diffusion models.

Findings

Estimator is unbiased and has finite variance.

Numerical tests confirm accuracy across multiple diffusion models.

Outperforms biased discretization methods.

Abstract

In this article we consider the development of unbiased estimators of the Hessian, of the log-likelihood function with respect to parameters, for partially observed diffusion processes. These processes arise in numerous applications, where such diffusions require derivative information, either through the Jacobian or Hessian matrix. As time-discretizations of diffusions induce a bias, we provide an unbiased estimator of the Hessian. This is based on using Girsanov's Theorem and randomization schemes developed through Mcleish [2011] and Rhee & Glynn [2015]. We demonstrate our developed estimator of the Hessian is unbiased, and one of finite variance. We numerically test and verify this by comparing the methodology here to that of a newly proposed particle filtering methodology. We test this on a range of diffusion models, which include different Ornstein--Uhlenbeck processes and the Fitzhugh--Nagumo model, arising in neuroscience.