Geometric approach to filtering some illustrations
Seminar Room 1, Newton Institute
Suppose we have an SDE on Rn+p which lies over an SDE on Rn for the natural projection of Rn+p to Rn. With some "cohesiveness" assumptions on the SDE on Rn we can decompose the SDE on the big space and so describe the conditional law of its solution given knowledge of its projection. The same holds for suitable SDE's on manifolds, and in some infinite dimensional examples arising from SPDE's and stochastic flows. The method also relates to a canonical decomposition of one diffusion operator lying over another. This approach will be illustrated by considering the conditional law of solutions of a simple evolutionary SPDE given the integral of the solution over the space variables, and by looking at the problem of conditioning a stochastic flow by knowledge of its one-point motion, with a related application to standard gradient estimates.
This is joint work taken from a monograph by myself, Yves LeJan, and Xue-Mei Li, The Geometry of Filtering to appear in Birkhauser's "Frontiers in Mathematics" series