What happens after a big insurance claim is paid on a rare diamond that's been stolen?

Let's say Peter Parker bought a huge, very unique diamond ring for Mary Jane Watson. He decided to insure it in case someone stole it and so he bought an insurance policy from Lloyds of London. Lo and behold someone stole Mary Jane's diamond and Lloyds of London paid off the insurance claim on the diamond. What happens to the diamond? Does Lloyds of London try to recover it? It's a big, unique diamond, and very expensive. Do they try to track the thing down? What's the likelihood that the thieves will be able to resell it and it and not suffer any consequences?