An Impulse Control Problem for a Stochastic PDE Arising in Non Linear Filtering.

Abstract

We consider the nonlinear filtering problem of a vector diffusion process, when several noisy vector observations with possibly different dimension of their range space are available. At each time any number of these observations (or sensors) can be utilized in the signal processing performed by the nonlinear filter. The problem considered is the optimal selection of a schedule of these sensors from the available set, so as to optimally estimate a function of the state at the final time. Optimality is measured by a combined performance measure that allocates penalties for errors in estimation, switching between sensor schedules and for running a sensor. The solution is obtained in the form of a system of quasi-variational inequalities in the space of solutions of certain Zakai equations.