Appl Math Optim 45:23–44 (2002)
2002 Springer-Verlag New York Inc.
An Approximation for the Zakai Equation
and J. Xiong
Department of Mathematics, University of Kansas,
405 Snow Hall, Lawrence, KS 66045-2142, USA
Department of Statistics, University of North Carolina,
Chapel Hill, NC 27599-3260, USA
Department of Mathematics, University of Tennessee,
Knoxville, TN 37996-1300, USA
Abstract. In this article we consider a polygonal approximation to the unnormal-
ized conditional measure of a ﬁltering problem, which is the solution of the Zakai
stochastic differential equation on measure space. An estimate of the convergence
rate based on a distance which is equivalent to the weak convergence topology is
derived. We also study the density of the unnormalized conditional measure, which
is the solution of the Zakai stochastic partial differential equation. An estimate of
the convergence rate is also given in this case.
Key Words. Filtering, Zakai equation, Stochastic differential equation, Weak
AMS Classiﬁcation. Primary 60G35, 93E11, Secondary 60F25, 60H10.
On the stochastic basis (, F, F
, P), let X be the d-dimensional signal process gov-
erned by the following stochastic differential equation (SDE):