Likely he is playing (on stairs) and also trying to control you by preventing the end of the petting session. He has trained you to do what he wants!
Use a water pistol or a can with pennies to rattle to discipline him if he bites. Have them ready. Set him up to train him. Dont pet him if he demands it. Dont pet him where he demands it. You choose when and where.
Choose a petting spot for during the training period where you are higher than him (chair, etc). Dont come to him by laying down on the floor to pet him, where he has the advantage. Make him come to you and begin and end that way. After 6 months you will likely be able to relax the rules as he will have developed an acceptable pattern of less aggressive play and you will have re-established yourself as higher in the households leadership hierarchy. Right now he believes you are his cat! You need to reverse that perception
From what I can tell, the behavior you describe is called "play aggression".
"The SF/SPCA Cat Behavior Department feels that play aggression in felines is a behavior issue that can be understood, improved, and lived with - as long as adopters understand the behavior and are willing to utilize suggested techniques consistently to ensure a
happy home for both person and kitty..." From the San Francisco SPCA under Resources - "Adopting a Play-Aggressive Cat", see sfspca.org
"Play Aggression" and "Play Therapy" are also explained on Dr. Jean Hofve's site LittleBigCat.com