Sarcasm is an adjusted stress reaction.

"Brave of him to say, when he really doesn't have much proof that the hostages are in fact alive."

I think he was lying through his teeth. He was lying to Amy in the hole, and he knew it, he was lying about the hostages being alive, and he knew it, and he was lying to the boy about everybody being safe, and he knew it.

By which I mean, he was talking about his hope, not about what he actually *knew* to be true.