A compact 380 or 9mm in a belly band sob position worked nicely for me. Larger guns tend to bounce around too much. And sob position has very little if any impact on your balance when jogging.
I don't jog so I can't add much input, but my buddy in the army says he wears a chest holster when he jogs and it stays pretty secure. I wouldn't want to sweat intensely with my gun on me though, so maybe consider another choice for defense if you're comfortable doing so.