There are numerous examples of other authors continuing the work of someone else. However, in the majority of situations, it's someone who is related to the original author and could be "deemed" to have inherited the rights to the story. I don't know if the situation was discussed prior, though -- I suspect it was. eg. Brian Herbert continuing Frank Herbert's "Dune" series. Also the recent new "Tolkein" story.
This is why I said it can be ethical for the story to be continued by another author, if the original author gave permission. I appreciate Snowy's view and I can't disagree with him, but he's using a lot tighter definition of "ethical" and "continuing" than I am. I'm looking at it more as completing something that was left unfinished, as a sort-of tribute to the original author.
My problem with the second question was the word "should", and that's why I answered "It depends". If someone asked me (as an example) to continue the work of a deceased author, I don't believe I "should". If I admired the deceased author's work, and their style was one that I was comfortable with AND if there were notes on how the original author had intended to the story to go, then I would seriously consider it. Unless all of those conditions were met, I would almost certainly politely decline.
I think I would be obliged to try to match the style of the original author if I was continuing their work, though it's possible I could slowly morph it towards my own style (if the continuation went long enough -- eg. a series of books). I remember one series I read where the first books were written in third person, and the last one was written in first person. That last one really grated with me -- it felt wrong. That's an example of how a change of style isn't appropriate (though in that particular case it was the same author all the way through -- I still don't know what they were thinking of....).
EDIT TO ADD:
There is a distinct difference between MIA and deceased. In the later case we know that the author is not going to continue the story, and there are probably kin who will know the author's wishes on the matter. In the former we don't have either of these. I would never consider continuing a story for someone who is MIA -- it wouldn't be right.