Discrete-space particle filters for reflecting diffusions

We consider the low observable filtering problem of detecting and tracking a target buried in high amplitude synthetic spatial observation noise. Motivated by fish farming applications, we constrain our target to live in a rectangular region, undergoing reflections at the boundary of this region, and moving in a manner described by the unique solution to a Skorohod stochastic differential equation. Observations are taken at discrete times and consist of a nonlinear partial function of the current state corrupted by additive noise. We use the reference probability method to describe the solution to this filtering problem in terms of a discrete-time version of the Duncan-Mortensen-Zakai equation and then use Markov chain approximations to produce an implementable approximate solution. The approximations incorporate discretizations of both space and amplitude directly into the unnormalized conditional distribution of the signal given the back observations. These approximations converge to the actual filtering conditional distribution as the discretization mesh is refined. The algorithm to implement our filter is reduced to an algorithm to implement a specific time-inhomogeneous Markov chain, which can be done using a single Poisson process and independent sequences of Bernoulli trials. The inhomogeneity is due to the observations themselves. The discretization of amplitude results in particles representing a small mass of the conditional distribution at particular grid points in the signal domain. These particles diffuse, drift, give birth, and die within the region similarly to those of continuous-state particle filters. The particles include information from the observations through observation-dependent births and deaths. We discuss issues like mean time to localize the target and fidelity of filter estimates at various signal to noise ratios, and give visual demonstrations of filter performance.