Yes, it makes no difference whether you say:
in the person's name
in the name of the person.
However, I will make two minor corrections, one of which may have just been a typo. First, in English, someone may purchase a ticket on the passenger's behalf, but the ticket must be in the passenger's name.
Second, It is unlikely (but not impossible) that you would have a fully paid ticket for an applicant. It is more likely that you would have a fully paid ticket for a passenger (eg when travelling on a plane or ship or possibly other forms of transport), or a customer (eg when attending the theatre, cinema, or a convention).