Unbiased Estimation of the Gradient of the Log-Likelihood for a Class of Continuous-Time State-Space Models

In this paper, we consider static parameter estimation for a class of continuous-time state-space models. Our goal is to obtain an unbiased estimate of the gradient of the log-likelihood (score function), which is an estimate that is unbiased even if the stochastic processes involved in the model must be discretized in time. To achieve this goal, we apply a \emph{doubly randomized scheme} (see, e.g.,~\cite{ub_mcmc, ub_grad}), that involves a novel coupled conditional particle filter (CCPF) on the second level of randomization \cite{jacob2}. Our novel estimate helps facilitate the application of gradient-based estimation algorithms, such as stochastic-gradient Langevin descent. We illustrate our methodology in the context of stochastic gradient descent (SGD) in several numerical examples and compare with the Rhee \& Glynn estimator \cite{rhee,vihola}.