Group invariance methods in nonlinear filtering of diffusion processes

Abstract

Given two "nonlinear filtering problems" described by the processes dx (t)i = fi (xi(t)) dt+gi (xi(t))dwi(t) i=1,2, dx (t)i = hi (xi(t)) dt+dvi(t), we define a notion of strong equivalence relating the solutions to the corresponding Mortensen-Zakai equations dui (t,x) = Lui i(t,x)dt + Li iui (t,x)dyt i, i=1,2, which allows solution of one problem to be obtained easily from solutions of the other. We give a geometric picture of this equivalence as a group of local transformations acting on manifolds of solutions. We then show that by knowing the full invariance group of the time invariant equations dui (t,x) = Lui i (t,x)dt, i=1,2, we can analyze strong equivalence for the filtering problems. In particular if the two time invariant parabolic operators are in the same orbit of the invariance group we can show strong equivalence for the filtering problems. As a result filtering problems are separated into equivalent classes which correspond to orbits of invariance groups of parabolic operators. As specific example we treat V. Bene¿'s case establishing from this point of view the necessity of the Riccati equation.