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In Crisan, Gaines and Lyons [SIAM J. Appl. Probab. 58 (1998) 313--342] we describe a branching particle algorithm that produces a particle approximation to the solution of the Zakai equation and find an upper bound for the rate of convergence of the mean square error. In this paper, the exact rate of convergence of the mean square error is deduced. Also, several variations of the branching algorithm with better rates of convergence are introduced.

Exact rates of convergeance for a branching particle approximation to the solution of the Zakai equation