Although Ian Mitchell asked some good questions, I'm more curious as to what you mean by "own". What does "ownership" of Jira, or any tool, actually entail? Also, why is the Product Owner not considered as a possible "owner" of Jira?
Why not whole Scrum Team? PO has some responsibility in managing backlog (which I guess would be stored in JIRA), while then the Dev Team should update status of their work items in JIRA. It is not too much of a hassle, and brings a lot of transparency. The SM makes couches the other roles to do as described in previous sentence.