  In this paper we prove a rate of convergence for the continuous time
filtering solution of a multiple timescale correlated nonlinear system to a
lower dimensional filtering equation in the limit of large timescale
separation. Correlation is assumed to occur between the slow signal and
observation processes. Convergence is almost sure in the weak topology. An
asymptotic expansion of the dual process for the solution to the Zakai
equation, and probabilistic representation using backward doubly stochastic
differential equations is leveraged to prove the result.
