Appl Math Optim (2014) 70:395–409
Filtering for Non-Markovian SDEs Involving Nonlinear
SPDEs and Backward Parabolic Equations
Daniela Ijacu · Marinela Marinescu
Published online: 27 March 2014
© Springer Science+Business Media New York 2014
Abstract We study a ﬁltering problem for non-Markovian SDE’s where the drift
vector ﬁelds commute with diffusion vector ﬁelds. The evolution of the conditioned
mean value will be decribed using a backward parabolic equation with parameters.
Keywords Non-Markovian SDE · Filtering · Backward parabolic equations
Mathematics Subject Classiﬁcation 60H15
The investigation of evolution equations with stochastic perturbations serves a large
variety of areas of applicability, mathematical ﬁnance as well.
Nonlinear SPDEs have applications in modelling of interest rates, in stochastic
control with partial information (as it is speciﬁed in Lions and Souganidis ) etc.
In  Marinescu and Varsan give representations for the conditioned mean under the
incomplete knowledge of the state variable which is the solution of a nonlinear sto-
chastic differential system. The authors deﬁne an admissible strategy in feedback form
associated with a ﬁnancial market. Other applications of SPDEs (including ﬁnance)
may be found in Da Prato and Tubaro  and El Karoui et al. .
In  Pardoux and Peng provide a probabilistic representation for the classical
solution of a system of quasilinear backward parabolic stochastic partial differential
equations via a system of “doubly stochastic” differential equations. In Buckdahn and
D. Ijacu (
) · M. Marinescu
Department of Applied Mathematics, The Bucharest University of Economics,
Calea Dorobantilor 15-17, 0105724 Bucharest, Romania