I have had my ileostomy for 10 months. I can jog, but it makes my stomach hurt right above my stoma. I usually stick to walking and maybe even a fast walk. I do have a hernia though and I am not sure if this is where the pain comes from.
I make sure to wear my hernia belt at all times while exercising.
I have also used a belly band that is made for pregnant women. That also helps to keep everything from jiggling around.