Thin-thick approach to martingale representations on progressively enlarged filtrations

We study the strong predictable representation property in the progressive enlargement F{τ} of a reference filtration F by a random time {τ}. Our approach is based on the decomposition of any random time in two parts, one overlapping F-stopping times (thin part) and the other that avoids them (thick part). We assume that the F-thin part of {τ} is nontrivial and prove in great generality a martingale representation theorem on F{τ}. The obtained result extends existing results in the literature, where the random time is usually considered equal to its thick part. We illustrate by some examples the case when F is the natural filtration of a L\'evy process.