Filtration shrinkage by level-crossings of a diffusion

Abstract

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 ∈ (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 j± i , which are related to the differential equation Au = λu.