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March 2007 Filtration shrinkage by level-crossings of a diffusion
A. Deniz Sezer
Ann. Probab. 35(2): 739-757 (March 2007). DOI: 10.1214/009117906000000683

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<⋯<xN 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 $\mathscr{A}$. 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 Fi, which are related to the differential equation $\mathscr{A}u=\lambda u$.

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A. Deniz Sezer. "Filtration shrinkage by level-crossings of a diffusion." Ann. Probab. 35 (2) 739 - 757, March 2007. https://doi.org/10.1214/009117906000000683

Information

Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1128.60070
MathSciNet: MR2308595
Digital Object Identifier: 10.1214/009117906000000683

Subjects:
Primary: 60G57 , 60J65
Secondary: 60G10

Keywords: characteristic measure , diffusion , point process of excursions , random measure , Regenerative sets

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 2 • March 2007
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