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.
In the current work we show that in the
case of dependent signal and (multidimensional) observation
there exists a version that is continuous in rough path metric.
This is joint work with
Dan Crisan (Imperial College London),
Peter Friz (TU Berlin) and
Harald Oberhauser (TU Berlin).