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Local martingale deflators for asset processes stopped at a default time $S^τ$ or right before $S^{τ-}$

Let $\mathbb{F}\subset \mathbb{G}$ be two filtrations and $S$ be a $\mathbb{F}$ semimartingale possessing a $\mathbb{F}$ local martingale deflator. Consider $τ$ a $\mathbb{G}$ stopping time. We study the problem whether $S^{τ-}$ or $S^τ$ can have $\mathbb{G}$ local martingale deflators. A suitable theoretical framework is set up in this paper, within which necessary/sufficient conditions for the problem to be solved have been proved. Under these conditions, we will construct $\mathbb{G}$ local martingale deflators for $S^{τ-}$ or for $S^τ$. Among others, it is proved that $\mathbb{G}$ local martingale deflators are multiples of $\mathbb{F}$ local martingale deflators, with a multiplicator coming from the multiplicative decomposition of the Azéma supermartingale of $τ$. The proofs of the necessary/sufficient conditions require various results to be established about Azéma supermartingale, about local martingale deflator, about filtration enlargement, which are interesting in themselves. Our study is based on a filtration enlargement setting. For applications, it is important to have a method to infer the existence of such setting from the knowledge of the market information. This question is discussed at the end of the paper.