In the case of low noise levels the optimal probability density function summarizing the available information about the state of a system can be accurately approximated by the product of a gaussian function and a linear function. The approximation preserves the ability to estimate to an accuracy of
O
(
λ
-2
) the expected value of any twice continuously differentiable function defined on the state space. The parameter
λ
depends on the noise level. If the noise level in the system is low then
λ
is large. A new filtering method based on this approximation is described. The approximating function is updated recursively as the system evolves with time, and as new measurements of the system state are obtained. The updates preserve the ability to estimate the expected values of functions to an accuracy of
O
(
λ
-2
). The new filter does not store previous measurements or previous approximations to the optimal probability density function. The new filter is called the asymptotic filter, because the definition of the filter and the analysis of its properties are based on the theory of asymptotic expansion of integrals of Laplace type. An analysis of the state propagation equations shows that the asymptotic filter performs better than a particular widely used suboptimal approximation to the optimal filter, the extended Kalman filter. The extended Kalman filter does not, in general, preserve the ability to estimate expected values to an accuracy of
O
(
λ
-2
). The computational cost of the asymptotic filter is comparable to that of the iterated extended Kalman filter.