I think the best way to ask this question is with an example.
We all know the server reset its dailies in midnight UTC, so, let's suppose it's the day of the lord 17 Jan 2012, 11:45 UTC, I'm with a group and we start Ascalonian Catacombs Dungeon doing the Path 1. But we just finish it at 18 Jan 2012, 00:15 UTC. The question is, for which day do this run counts? Does it count as a daily run for 17 january or 18 january? In its beggining or end?