Unique characterization of conditional distributions in nonlinear filtering
Let X be a ``signal'' of interest, unobservable, and Y be an observable process related to X. The aim of the paper is to present a unique characterization of the law \(\pi\) of X given Y for couples (X,Y) solving a martingale problem in the sense that \[ (M):f(Z(t))- \int^{t}_{0}Af(Z(s))ds \] is a martingale for f in the domain of a fixed operator A, \(Z=(X,Y)\). The solution is unique if all solutions have the same finite dimensional distributions. The method consists in introducing an auxiliary problem, the filtered martingale problem, which has the form \[ (FM):\mu_ tf(\cdot,U(t))- \int^{t}_{0}\mu_ sAf(\cdot,U(s))ds \] (\(\mu\) a probability measure). One recognizes that \(\mu\) could be \(\pi\). One then introduces conditions on (M) to insure that \(\pi\) is the only solution of (FM). A number of uniqueness theorems are derived and it is shown that the method improves on many of the similar results known so far.
scientific article; zbMATH DE number 4069937