I don't understand how to speedrun this game because on ps3 you can't record the game and therefore you can't have a proof to submit your speedrun So if someone knows how to do it, then his answer is welcome here! Thanks in advance.

You'll need a capture card and a recording software like OBS to record gameplay for PS3, 360 too. Or you could just point your webcam/camera at your TV, that works as well. Just as long as it's decent quality and actually watchable.