Appl Math Optim 42:315–332 (2000)
2000 Springer-Verlag New York Inc.
A Direct Approach to Deriving Filtering Equations
for Diffusion Processes
N. V. Krylov and A. Zatezalo
127 Vincent Hall, University of Minnesota,
Minneapolis, MN 55455, USA
Abstract. Filtering equations are derived for conditional probability density func-
tions in case of partially observable diffusion processes by using results and methods
from the L
-theory of SPDEs. The method of derivation is new and does not require
any knowledge of ﬁltering theory.
AMS Classiﬁcation. Primary 60G35, Secondary 62M20.
Key Words. Filtering, Diffusion processes, L
-Theory of SPDEs.
The main purpose of the article is to give a derivation of the so-called ﬁltering equation
for partially observable diffusion processes on the basis of the L
-theory of SPDEs from
 without using any facts from ﬁltering theory itself. We refer the reader to  for a
standard derivation, bibliography, and historical remarks.
As in , where ﬁnite-state Markov processes are considered, we follow a “direct”
PDE approach as opposed to a purely probabilistic one. To give an idea of what we mean
by these approaches, we consider Itˆo’s stochastic equation
say in one dimension with nonrandom coefﬁcients satisfying appropriate conditions.
(x) be a solution of this equation starting at x. Take a smooth bounded function
φ(x) and deﬁne v(t, x) = Eφ(x
(x)). Then under appropriate conditions v satisﬁes
The work of the ﬁrst author was partially supported by NSF Grant DMS-9625483.