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On lower bounds for the nonlinear filtering problem

The system model dx_{t}=m(x_{t})dt+\sigma_{o}dw_{t} and the observation dy_{t}=h(x_{t})dt+\rho dv_{t} are considered, and a lower bound on the error for the least-square estimate of a function of x_{t} , say z(x_{t}) , is derived. This bound is then applied to derive a lower bound on the error in estimating x_{t} in the case where the constant \sigma_{o} in the equation for the system dynamics is replaced by \sigma (x_{t}) .