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<⋯N in ℝ, the region indicator function R(x) assumes the value i if x∈(xi−1, xi]. We take $\mathbb{F}$ to be the filtration generated by (R(Xt))t≥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 Lévy measures Fj±i, which are related to the differential equation $\mathcal{A}u=\lambda u$ .
Publié le : 2007-03-14
Classification:
Random measure,
diffusion,
point process of excursions,
characteristic measure,
regenerative sets,
60J65,
60G57,
60G10
@article{1175287761,
author = {Sezer, A. Deniz},
title = {Filtration shrinkage by level-crossings of a diffusion},
journal = {Ann. Probab.},
volume = {35},
number = {1},
year = {2007},
pages = { 739-757},
language = {en},
url = {http://dml.mathdoc.fr/item/1175287761}
}
Sezer, A. Deniz. Filtration shrinkage by level-crossings of a diffusion. Ann. Probab., Tome 35 (2007) no. 1, pp. 739-757. http://gdmltest.u-ga.fr/item/1175287761/