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The solution of the continuous time filtering problem can be represented as aratio of two expectations of certain functionals of the signal process that areparametrized by the observation path. We introduce a new time discretisation ofthese functionals corresponding to a chosen partition of the time interval andshow that the convergence rate of discretisation is proportional with thesquare of the mesh of the partition.

A second order time discretization of the solution of the non-linear filtering problem