Approximate MMSE and MAP estimation using continuous-time particle filter

The algorithm is proposed for solving approximately the optimal filtering problem for nonlinear continuous-time stochastic observation systems that provides two estimates for the state. These estimates are the minimum mean squared error estimate and the maximum a posteriori estimate. The proposed algorithm is based on the continuous-time particle filter, which corresponds to the Duncan–Mortensen–Zakai equation. To find the mode of the conditional distribution approximately, the Edgeworth series is used for the conditional probability density function expansion. This approach allows one to significantly reduce the computation time in contrast to finding the mode by estimating the conditional probability density function, for example, by the histogram or the kernel density estimation.