How can I upload media without setting directory permissions to 777?

I am having a problem with a new WordPress installation that I have not experienced before. I cannot upload media (or make a backup using BackWPup for instance) unless I chmod the parent directory for the new files to 777 (which I believe is insecure). As I understand it this means that WordPress is not running as Owner or Group user as it should be. The FTP account that I used to install is the ‘Owner’ account and the host company say that it has full permissions. Is there something I should be doing to somehow make WordPress run as the Owner?