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

Unbiased Estimation of the Hessian for Partially Observed Diffusions