The partial malliavin calculus and its application to non-linear filtering

In this article, the Malliavin calculus is used to derive regularity properties of the conditional distribution of one ltd process given a second Ito process. The relation between the processes involved is the usual one assumed in the study of filtering theory. The non-degeneracy which we require is stated in terms of Malliavins covariance matrix in Theorem (3.15). More practical conditions are given in Lemma (3.19) for general Tto processes and in Lemma (3.29) for diffusions. Finally, in Theorem (4.6) a “localized” version of these results is given for diffusions.