The movies do not explain this, but based on the hand-to-hand combat scenes from the Bourne movies, which discipline(s) does he seem to be skilled?
The Bourne Directory says:
Jason Bourne is a masterful expert in Kali martial arts. He is seen using it in the movies and video game.
Further, amongst his weapons are listed are:
- Jeet Kune Do
- Krav Maga
- Kali Martial Arts
Finally, the page about the game Robert Ludlum's The Bourne Conspiracy says:
As a third-person espionage action game, players move from fistfights to firefights to car chases to experience sequences that aim to seamlessly blend Bourne's signature Kali martial arts with intense shooting and the ability to escape and evade deadly situations.
I believe it's a Hollywood blend based on a Filipino style called Kali but I'm afraid I don't have any supporting links, do seem to remember something in one of the directors commentaries (or a extra) from Identity?