Did it turn at all before it broke off? If so, that's a help. If not, get ready for it to be a PITA.
Is part of the stud sticking out the other side? If so, then try what I said before on that side.
If not, try one of these two, depending on what tools you have access to:
1. (Probably better of the two) drill a hole as mentioned in other posts. If the hole is threaded all the way through, you may get lucky and it will thread itself out.if not threaded all the way through, and you don't mind spending a few bucks, get an appropriately sized, left handed drill bit. Drill pilot hole with regular drill bit, then use left handed. Again, maybe you get lucky and it threads itself out. Don't know what kind of close quarters you are working with, but a right angle drill may be of extreme help.
2. Get a small chisel and chisel phillips head (cross) into bolt. Get a hammer driver with phillips head bit on it, and drive the bolt out or through, depending on if threaded through or not, as above.
Hope this helps, keep us posted.