Appl Math Optim 50:119–134 (2004)
2004 Springer-Verlag New York, LLC
Explicit Solution of a Non-Linear Filtering Problem for
L´evy Processes with Application to Finance
Thilo Meyer-Brandis and Frank Proske
Centre of Mathematics for Applications (CMA),
Department of Mathematics, University of Oslo,
P.O. Box 1053 Blindern, N-0316 Oslo, Norway
Abstract. In this paper we explicitly solve a non-linear ﬁltering problem with
mixed observations, modelled by a Brownian motion and a generalized Cox process,
whose jump intensity is given in terms of a L´evy measure. Motivated by empirical
observations of R. Cont and P. Tankov we propose a model for ﬁnancial assets,
which captures the phenomenon of time inhomogeneity of the jump size density.
We apply the explicit formula to obtain the optimal ﬁlter for the corresponding
Key Words. L´evy processes, Non-linear ﬁltering, Stochastic partial differential
equations, White noise analysis, Mathematical ﬁnance.
AMS Classiﬁcation. 60G51, 60G35, 60H15, 60H40, 60H15, 91B70.
In this paper we derive an explicit solution for a non-linear ﬁltering problem for L´evy
We are interested in the following non-linear ﬁltering model:
We consider a process X
, which follows the dynamic of the stochastic differential
) dt + σ(X