Appl Math Optim 42:51–71 (2000)
2000 Springer-Verlag New York Inc.
Robust Approximation in a Filtering Problem
with Real State Space and Counting Observations
and G. Nappo
Dipartimento di Matematica, Universit´a di Roma “Tor Vergata,”
Via della Ricerca Scientiﬁca, 00133 Roma, Italia
Dipartimento di Matematica, Universit´a di Roma “La Sapienza,”
Piazzale A. Moro 2, 00185 Roma, Italia
Communicated by D. Ocone
Abstract. Let (X
) be a pure jump Markov process, where X
takes values in
R and Y
is a counting process. We compare the ﬁlter of this system and a ﬁlter of
a suitably modiﬁed system. We compute an explicit bound for the distance in the
so-called bounded Lipschitz metric between the two ﬁlters. Finally we show how
to use this bound to construct a discrete space approximation of the ﬁlter.
Key Words. Filtering, Counting process, Jump Markov process, Coupling.
AMS Classiﬁcation. 60G35, 60J75, 60F99.
The problem of robust ﬁlter approximation arises naturally in nonlinear ﬁltering theory,
and we may quote  as a ﬁrst contribution on this problem. Among the others, we
mention , where this problem is treated from a Bayesian point of view for a general
diffusion model and an error bound is given. As far as the approximation problem alone
is concerned we quote the recent paper , where the problem is treated with a functional
series approach, again for a diffusion model and again an explicit error bound is given.
This paper is also an excellent source of references.
This research was partially supported by MURST.