For the sake of questioning, let's use the example of Tom Hanks from Cast Away. An American citizen survives the crash of a private commerical airplane in the middle of nowhere and becomes stranded on a previously undiscovered island.
After years of searching and thousands of man hours spent searching for him he's one day found by a passing cruise ship. Tom says "No thanks" to the lift, but it is still reported to the authories and to his loved one that he's having the time of his life on his own beach with Wilson.
Is he still recquired to come back and sign off on all the stuff that previously reported him as dead? Are the search and rescue forces owed a debt now for the time and money spent looking for him? Does being the first one on the island make it his legally to reside on for the rest of his days?
Tom -> <-pretend that's a volleyball