A Problem in Stochastic Averaging of Nonlinear Filters

Abstract

We consider the stochastic averaging of a particular degenerate problem in nonlinear filtering. In particular, we consider a problem on the cylinder with fast angular diffusion and slow axial diffusion. The observation process depends only on the fast angle. We show that as the separation of scales (between fast and slow) grows, one can approximate the filter for the original problem by a much simpler one. Although our problem is similar to a number of others considered in the literature, the techniques required are different from the known ones.

Filtering the Maximum Likelihood for Multiscale Problems

“…We note that results similar to Theorem 3.1 appear elsewhere in the literature, e.g., Kleptsina et al (1997), Ichihara (2004), Park et al (2008Park et al ( , 2011Park et al ( , 2010, Imkeller et al (2013), but with slightly different assumptions and setup. The main difference is that Theorem 3.1, when compared to the previous works, states the convergence result under the measure parameterized by the true parameter value (i.e., the measure under which the observations are made, where θ = α) with the filters converging for any parameter value.…”

“…In subsection 3.1 we use the convergence results found in Imkeller et al (2013) (see also Park et al (2008Park et al ( , 2011Park et al ( , 2010, Imkeller et al (2013)) to prove convergence in probability of the filter for a class of unbounded test functions (e.g., for the eigenfunctions of the operator L θ ). Then, in subsection 3.2 we will use these results to derive a CLT for the log-likelihood function, which is the main result of the paper.…”

“…Proof of Theorem 3.1. The proof of Theorem 3.1 follows by the results of Imkeller et al (2013) (see also Park et al (2008Park et al ( , 2011Park et al ( , 2010) after we adjust for the parameter mismatch. In particular, the main difference that Theorem 3.1 has when compared to the previous works is that under the measure parameterized by the true parameter value (i.e., the measure under which the observations are made) the filters will converge for any parameter value.…”

Efficient nonlinear filtering of a singularly perturbed stochastic hybrid system

“…† By means of various coordinate changes, we can transform the problem so that dW can have any positivedefinite covariance matrix, which may, in fact, depend on Θ εThis type of result has been covered in the literature within the framework of homogenization theory; we refer to [16] and the references therein for more detail. Methodologically, this study is similar to [15] and [16]. In the former work, the observation becomes independent of the system in the limit, while the latter has an explicit dependence on the slow variable in the limit.…”

“…In the former work, the observation becomes independent of the system in the limit, while the latter has an explicit dependence on the slow variable in the limit. In [15,16], the fast motion was a fast angular drift. In contrast to these papers, the fast motion (1.1) has both drift and diffusion, so the speed of averaging depends on a spectral gap.…”