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 x₁<⋯<x_N in ℝ, the region indicator function R(x) assumes the value i if x∈(x_i−1, x_i]. We take $\mathbb{F}$ to be the filtration generated by (R(X_t))_t≥0, where X is a diffusion with infinitesimal generator $\mathscr{A}$. We prove a martingale representation theorem for $\mathbb{F}$ in terms of stochastic integrals with respect to N random measures whose compensators have a simple form given in terms of certain Lévy measures F^j±_i, which are related to the differential equation $\mathscr{A}u=\lambda u$.