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 $x_1<\mathrm{.}\mathrm{.}\mathrm{.}<x_N$ in $\mathbb{R}$, the region indicator function $R(x)$ assumes the value $i$ if $x\in (x_{i-1},x_i]$. We take $\mathbb{F}$ to be the filtration generated by $(R(X_t){)}_{t\ge 0}$, where $X$ is a diffusion with infinitesimal generator $\mathcal{A}$. 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 L'{e}vy measures $F_i^{j\pm }$, which are related to the differential equation $\mathcal{A}u=\lambda u$.