On lower bounds for the nonlinear filtering problem (Corresp.)

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})
.