Filtering for piecewise linear drift and observation

The filtering problem for piecewise linear drift and observation functions is reduced to an initial-boundary value problem. The "corners" give rise to local time terms. A finite number of sufficient statistics appear, in the form of the values and one-sided derivatives of the conditional density at the "corners", or more generally in the form of weights in a representation of the conditional density by potentials. Both kinds of statistics propagate according to linear Volterra equations, and must be considered as infinite-dimensional. The theory developed here for piecewise linear dynamics enhances the study of the general nonlinear filtering problem in a natural way: Nonlinear functions can be approximated over bounded intervals by polygons, to any degree of accuracy; by constructing or calculating the optimal filter for the approximating piecewise linear dynamics as indicated in this paper, one can conceivably obtain very good sub-optimal filters for general nonlinear dynamics. That the results extend to many dimensions is far from clear, but likely whenever the necessary local times can be defined.