## Accurate Evaluation of Stochastic Wiener Integrals with Applications to Scattering and Nonlinear Filtering Problems

**Title**: Accurate Evaluation of Stochastic Wiener Integrals with Applications to Scattering and Nonlinear Filtering Problems

**Authors**:

Hopkins, William E

Baras, John, S.

**Conference**: Recent Developments in Control Theory and its Applications pp. 54-68

**Date**: December 31 - December 31, 1982

Using s one approximation formulas for stochastic Wiener function Space integrals , it is possible to approximate the conditional densities which arise in the nonlinear filtering of diffusion processes to within O(n), with n >= 1 arbitrary, by n-fold ordinary integrals. The latter have the simple form of a ‘rectangular rule”, but their accuracy is an order of magnitude better . Then – fold integral. can be further decomposed into a recursion involving in one dimensional integrals. The sequence is recursive in the increments of the observation process in the filtering problem . It is not, however , recursive in time. The one dimensional integrals are naturally treated by an m – step Gaussian quadrature which has an error proportiona1 to nm! / 2″ (2 m) : ). (The proportionality constant can be estimated and optimized. ) The computation of these individual integrals can be reduced further by exploiting certain inherent symmetries of the problem, and by doing some preliminary, ‘off-line’ computing. The end result is a highly accurate, computationally efficient numerical algorithm for evaluating conditional densities for a substantial class of non linear filtering problems . By accepting light reductions in accuracy , One can Obtain an algorithm (apparently) fast enough, when efficiently coded, for “Online” recursive filtering in real-time