Nonlinear filtering with signal dependent observation noise
Dan Crisan, Imperial College
Michael A. Kouritzin, University of Alberta
Jie Xiong, University of Kentucky
Abstract
The paper studies the filtering problem for a non-classical frame-
work: we assume that the observation equation is driven by a signal
dependent noise. We show that the support of the conditional distri-
bution of the signal is on the corresponding level set of the derivative of
the quadratic variation process. Depending on the intrinsic dimension
of the noise, we distinguish two cases: In the first case, the conditional
distribution has discrete support and we deduce an explicit represen-
tation for the conditional distribution. In the second case, the filtering
problem is equivalent to a classical one defined on a manifold and we
deduce the evolution equation of the conditional distribution. The re-
sults are applied to the filtering problem where the observation noise
is an Ornstein-Uhlenbeck process.
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