I have been having this issue off and on since I first installed Windows Phone 8.1 last May through the Preview for Developers program. At one point, I think it was resolved with an intermediate update, but I've been dealing with this same issue for months now. I'm currently running the preview of WP 8.1 update 1 on my Lumia 920 since it has not yet officially been released by AT&T.
Per several threads I just read, I have allowed Calendar to run in the background even when Battery Saver is on. I doubt that will make a difference since the app shows 0% use and 0% in the background. The only way I can get the lock screen to update to the current day's calendar events is to power cycle the phone or soft reset. Simply changing the detailed app on the lockscreen to a different app then back to the calendar didn't help. All it did was cycle to the next calendar event for the day it was stuck on (2 days ago). I tried again to see if it would advance to the next calendar event, but it was still stuck on the last calendar event for 2 days ago.