Progressive Enlargement of Filtrations with Initial Times

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What happens after a default: The conditional density approach (2009, result): “a result established in [ 10 ], simplified by using the fact that any F-martingale is continuous at”
Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management (2010, background): “Now, since the semimartingale property is preserved under the density hypothesis for this progressive enlargement of filtration, (see [10]), the process W remains a semimartingale under (P,G) (with a canonical decomposition, which can be explicitly expressed in terms of the density).”