  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 doubly randomized scheme,
that involves a novel coupled conditional particle filter (CCPF) on the second
level of randomization. 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.
