I haven't had this problem, but if you goto the Start screen and search for "pen and touch" under settings, there should be a way to reset the Pen settings. I'd try re-calibrating the pen, and seeing if those settings might affect something. It could certainly be a faulty pen, but I'd try messing with settings first.
I don't think the Pen has a battery or any kind of power, but then again I'm attaching my Pen to the AC magnet when it's not in use (and not plugged in). Maybe try that too?