  We consider the problem of statistical inference for a class of
partially-observed diffusion processes, with discretely-observed data and
finite-dimensional parameters. We construct unbiased estimators of the score
function, i.e. the gradient of the log-likelihood function with respect to
parameters, with no time-discretization bias. These estimators can be
straightforwardly employed within stochastic gradient methods to perform
maximum likelihood estimation or Bayesian inference. As our proposed
methodology only requires access to a time-discretization scheme such as the
Euler-Maruyama method, it is applicable to a wide class of diffusion processes
and observation models. Our approach is based on a representation of the score
as a smoothing expectation using Girsanov theorem, and a novel adaptation of
the randomization schemes developed in Mcleish [2011], Rhee and Glynn [2015],
Jacob et al. [2020a]. This allows one to remove the time-discretization bias
and burn-in bias when computing smoothing expectations using the conditional
particle filter of Andrieu et al. [2010]. Central to our approach is the
development of new couplings of multiple conditional particle filters. We prove
under assumptions that our estimators are unbiased and have finite variance.
The methodology is illustrated on several challenging applications from
population ecology and neuroscience.
