Progressive enlargements of filtrations with pseudo-honest times

Abstract

We deal with various alternative decompositions of $\mathbb{F}$-martingales with respect to the filtration $\mathbb{G}$, which represents the enlargement of a filtration $\mathbb{F}$ by a progressive flow of observations of a random time that either belongs to the class of pseudo-honest times or satisfies the extended density hypothesis. Several related results from the existing literature are revisited and essentially extended. Results on $\mathbb{G}$-semimartingale decompositions of $\mathbb{F}$-local martingales are crucial for applications in financial mathematics, most notably in the context of credit risk modeling and the study of insider trading where the enlarged filtration plays a vital role. We outline potential applications of our results to problems arising in financial mathematics.