We always register it under your name. You will be paying for the domain, so you will have complete ownership.
Only nonpayment will prevent you from having complete control of your domain. Should you fail to make payment, Mr.Host will retain control of the domain — including the right to transfer, sell, or modify it — until your balance is paid. Virtually all registrars operate this way.
As long as your payment is current, you will have complete control of your domain.