Filtration shrinkage by level-crossings of a diffusion

We develop the mathematics of a filtration shrinkage model that has recently been considered in the credit risk modeling literature. Given a finite collection of points x(1) < ... < x(N) in R, the region indicator function R(x) assumes the value i if x is an element of (x(i - 1), x(i)]. We take F to be the filtration generated by (R(X-t))(t >= 0), where X is a diffusion with infinitesimal generator A. We prove a martingale representation theorem for F in terms of stochastic integrals with respect to N random measures whose compensators have a simple form given in terms of certain Levy measures F-i(j +/-) which are related to the differential equation Au = lambda u.