No.
You can, however, tie up your TSO session by continuously checking (probably via STATUS) the condition of some JOB until it's finished.
Easier, if you plan ahead, is to have the job-in-question send a message to a VM disconnected service machine which is prepared to react to that message. A DSC on VM is the nearest thing an application programmer will get to having his own started task (which is what the original question asks for).
There are packages (CA-7 for example) which do this function on MVS.