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The optimal nonlinear filtering of certain vector-valued diffusion processes embedded in white noise is considered. We derive upper and lower bounds on the minimal causal mean-square error. The derivation of the lower bound is based on information-theoretic considerations, namely the rate-distortion function ( -entropy). The upper bounds are based on linear-filtering arguments. It is demonstrated that for a wide class of high-precision systems, the upper and lower bounds are tight within a factor of 2 or better.