Have had unknown long distance telephone charges for calls we did not make. Cox refuses to acknowledge they have a problem. What can I do.

Permission?  Permission from whom?  Your question begs the question:  who else...other than we...is in your house?  Just because you didn't permit the calls, it doesn't mean they weren't dialed.

What's the dates and times of the calls?  Who was home during those dates and times?  Do you have a home-security system.  Security systems routinely "call" their mother ship with status updates.  Those would be easy to identify because they're chronologically scheduled.

Why does this keep happening?  Maybe because it's easy to do in your home.