Filtration shrinkage by level-crossings of a diffusion

TL;DR

This paper develops a mathematical framework for filtration shrinkage models driven by diffusion processes and level-crossings, with applications in credit risk modeling, providing a martingale representation theorem in this context.

Contribution

It introduces a new filtration shrinkage model based on diffusion level-crossings and proves a martingale representation theorem for it, linking stochastic integrals to Lévy measures.

Findings

Martingale representation theorem for the filtration

Explicit form of compensators using Lévy measures

Application to credit risk modeling

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 $\mathbb{R}$, 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 $(R(X_t))_{t\geq0}$, 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 $F^{j\pm}_i$, which are related to the differential equation $\mathcal{A}u=\lambda u$.