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Thin-thick approach to martingale representations on progressively enlarged filtrations (2406.08983v2)
Published 13 Jun 2024 in math.PR
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.