[taijitsu]

Hi,
I'm new here and practice Tai-jitsu (name given to that martial art)
Now I'm very confuse of spelling and need tha help of a japanese to clarified that.
In U.S jitsu is called jutsu (u not i) and in spain, france, europe it's jitsu (with i) so if you go in america you'll say ju-jutsu (the i of ju is also removed to make ju in U.S), ken-jutsu... in france it's jiu-jitsu, ken-jitsu.

So my question is what is the correct spelling, why do they differs? is it because english doesn't know how to say jiu or jit? how do you say in japanese? will it have the "i" or not?

Thank you