  We develop a (nearly) unbiased particle filtering algorithm for a specific
class of continuous-time state-space models, such that (a) the latent process
$X_t$ is a linear Gaussian diffusion; and (b) the observations arise from a
Poisson process with intensity $\lambda(X_t)$. The likelihood of the posterior
probability density function of the latent process includes an intractable path
integral. Our algorithm relies on Poisson estimates which approximate
unbiasedly this integral. We show how we can tune these Poisson estimates to
ensure that, with large probability, all but a few of the estimates generated
by the algorithm are positive. Then replacing the negative estimates by zero
leads to a much smaller bias than what would obtain through discretisation. We
quantify the probability of negative estimates for certain special cases and
show that our particle filter is effectively unbiased. We apply our method to a
challenging 3D single molecule tracking example with a Born and Wolf
observation model.
