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

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