  The solution of the continuous time filtering problem can be represented as a
ratio of two expectations of certain functionals of the signal process that are
parametrized by the observation path. We introduce a class of discretization
schemes of these functionals of arbitrary order. The result generalizes the
classical work of Picard, who introduced first order discretizations to the
filtering functionals. For a given time interval partition, we construct
discretization schemes with convergence rates that are proportional with the
$m$-power of the mesh of the partition for arbitrary $m\in\mathbb{N}$. The
result paves the way for constructing high order numerical approximation for
the solution of the filtering problem.
