  Let $\mathbb{F}\subset \mathbb{G}$ be two filtrations and $S$ be a
$\mathbb{F}$ semimartingale possessing a $\mathbb{F}$ local martingale
deflator. Consider $\tau$ a $\mathbb{G}$ stopping time. We study the problem
whether $S^{\tau-}$ or $S^{\tau}$ 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 $S^{\tau-}$ or for $S^{\tau}$. 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 Az\'ema supermartingale of $\tau$. The proofs of the
necessary/sufficient conditions require various results to be established about
Az\'ema 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.
