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.
Highlights
Let (Ω, P) be a probability space on which we have defined a homogeneous Markov process X .We can obtain information on X by observing an associated process y which is a function of X plus random noise n: yt = h(X t ) + nt . (1.1)In most of the literature nt is modeled by white noise, which does not exist in the ordinary sense, but rather as a distribution of a generalized derivative of a Brownian motion
4 cos x sin x is a function of aσ1 so (IN1) holds and the level sets Mz are described by Mz = {x i(z) : i ∈ Z}, where Z denotes the collection of integers and the continuous functions x i : R3 → R are defined as x i(z) = x i(z1, z2, z3) =
As we indicated in the introduction, the OU-process is an approximation of the white noise which exists in the sense of generalized function only
Summary
Let (Ω, , P) be a probability space on which we have defined a homogeneous Markov process X. The authors assume that the map t → h(X t ) is differentiable To remove this restrictive condition, Bhatt and Karandikar [1] consider the variant observation model t yt = α h(Xs)ds + Ot ( t −α−1 )∨0 for α > 0 and obtain the same results for this modified model. In [7], they study the filtering problem with perfect observations That is, in their set-up, the observation process Y is a deterministic function of the signal X. We show that the filtering problem that we are interested in can be reduced to one where the observation process has two components: one that is perfect (in the language of Ioannides and LeGland ) and one that is of the classical form (see Lemma 2.3 below)
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