The filtering problem is concerned
with the distribution of an Ito diffusion (the signal)
conditioned on another process (the observation).
In the case of independent signal and observation it was shown in Clark-Crisan
[On a robust version of the integral representation formula of nonlinear filtering.
Probability Theory and Related Fields, 133, 2005.]
that there exist a version of the conditional distribution that depends
continuously (in supremem norm) on the observation process.
Kushner [A robust discrete state approximation to the optimal nonlinear filter for a diffusion. Stochastics, 1980.]
showed that a certain approximation to the conditional distribution is also continuous in the signal,
_uniformly_ in the approximation parameter.
We extend this result to different approximations, including one using the cubature method on Wiener space.
In the case of dependent signal and (multidimensional) observation we show
there exists a version of the conditional distribution that is continuous in rough path metric.
The question of robust approximations in this case is still mainly open.
This is joint work with
Dan Crisan (Imperial College London),
Peter Friz (TU Berlin) and
Harald Oberhauser (TU Berlin).