Thin-thick approach to martingale representations on progressively enlarged filtrations

TL;DR

This paper develops a new martingale representation theorem for progressively enlarged filtrations by decomposing a random time into parts that either overlap or avoid the original filtration's stopping times, extending previous results.

Contribution

It introduces a thin-thick decomposition of random times and proves a martingale representation theorem in the enlarged filtration, broadening the scope of prior work.

Findings

Established a martingale representation theorem for the thin part of the random time.

Extended results to include random times that do not avoid the original filtration.

Provided applications to the natural filtration of Lévy processes.

Abstract

We study the predictable representation property in the progressive enlargement F^\tau of a reference filtration F by a random time \tau. Our approach is based on the decomposition of any random time into two parts, one overlapping F-stopping times (thin part) and the other one that avoids F-stopping times (thick part). We assume that the F-thin part of \tau is nontrivial and prove a martingale representation theorem on F^\tau. We thus extend previous results dealing with F-avoiding random times. We collect some examples of application to the enlargement of the natural filtration of a L\'evy process.