Accurate Evaluation of Conditional Densities in Nonlinear Filtering

Title : Accurate Evaluation of Conditional Densities in Nonlinear Filtering
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Conference : Advances in Filtering and Stochastic Control Vol. 42, pp. 54-68

Using some 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 0(n-2), with n􀀑l 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. The n-fold integral can be further decom- posed into a recursion involving n 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 proportional to nm!/[2m(2m)!]. (The proportionality constant can be estimated and optimized.) The computa- tion of these individual integrals can be reduced further by exploiting certain inherent symmetries of the problem, and by doing some prelim- inary, “off-line” computing. The end result is a highly accurate, computationally efficient numerical algorithm for evaluating condi­tional densities for a substantial class of nonlinear filtering problems. By accepting slight reductions· in accuracy, one can obtain an algorithm (apparently) fast enough, when efficiently coded, for “on-line,” recursive filtering in real time.

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