The way to find this answer is to look at chapter 11.1 of the Defense Acquisition Guidebook. It contains all the key information you need on leading and managing Joint Programs. Here is the link.
https://acc.dau.mil/CommunityBrowser.aspx?id=488720 . The Milestone Decision Authority makes the call on which service will serve as the "Executive Agent" and leads the Joint Program.
Considering the recommendation of the Joint Staff and the Heads of the DoD Components, the Milestone Decision Authority decides whether to place the program under joint acquisition management. The Milestone Decision Authority should make this decision and, if appropriate, designate the Lead Executive DoD Component, as early as possible in the acquisition process.
The lead Executive DoD Component (the service) has several responsibilities such as funding, leadership, lead Operational Test Agency, etc. Again, see chapter 11 of the DAG for further information.