How to check who completed a task

I have a case where I have some user tasks which are marked complete using REST calls. Is there any way I can check later who marked the task as done?

I see that assigned is not changed in the task.

The task should first be claimed by the user which makes it the assignee. The assignee can, after task completion, be found in the assignee field of the historic task instance.

In that case should not the REST call for complete stop the user and check if the desired fields are set or set them

A task can go unassigned and then be completed. In this case there would be no assignee in the historic task instance.