Appl Math Optim 39:327–335 (1999)
1999 Springer-Verlag New York Inc.
A Support Theorem for the Filter under
Inﬁnite Dimensional Noise and
Unbounded Observation Coefﬁcients
D´epartement de Math´ematiques, Universit´e de Metz,
BP 80794, F-57012 Metz Cedex, France
Communicated by M. Zakai
Abstract. In this paper we consider a nonlinear ﬁltering problem with an un-
bounded observation coefﬁcient, correlated noises, and a signal process driven by
an inﬁnite dimensional Brownian motion. We prove that the unnormalized ﬁlter
admits a smooth density which is in the Schwartz space and we give a description
of the support of the law of this density.
Key Words. Support theorem, Nonlinear ﬁltering, Stochastic differential equa-
AMS Classiﬁcation. 60G35, 60H10.
The purpose of this paper is to describe the support of the law of the density of the
unnormalized ﬁlter associated with a correlated nonlinear ﬁltering problem with inﬁnite
dimensional noises and an unbounded observation coefﬁcient, by using a set of solutions
of a controlled system.
A description of the law of a stochastic process, deﬁned as a solution of a stochastic
differential equation, as the closure of a set of trajectories of a controlled system deduced
from the original stochastic system by substituting the Brownian motion by an H
control, has been initiated by the celebrated paper of Stroock and Varadhan .
Assuming that the law of the unnormalized ﬁlter asociated with a nonlinear ﬁltering
problem with independent noises has a density with respect to the Lebesgue measure,