  We consider the problem of parameter estimation for a class of
continuous-time state space models. In particular, we explore the case of a
partially observed diffusion, with data also arriving according to a diffusion
process. Based upon a standard identity of the score function, we consider two
particle filter based methodologies to estimate the score function. Both
methods rely on an online estimation algorithm for the score function of
$\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. The
first approach employs a simple Euler discretization and standard particle
smoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time,
where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step.
The second approach is new and based upon a novel diffusion bridge
construction. It yields a new backward type Feynman-Kac formula in
continuous-time for the score function and is presented along with a particle
method for its approximation. Considering a time-discretization, the cost is
$\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs,
we then consider multilevel methodologies for the score function. We illustrate
our parameter estimation method via stochastic gradient approaches in several
numerical examples.
