Some of the others have mentioned it but I just wanted to emphasize that when you're jumped from a position of advantage turning into the threat is usually the best option. Do not make the mistake that many in the AH arenas seem to make and that's equating a turn into the threat with a HO. Turning into the threat is not a "HO" and is the very first thing a new fighter pilot learns to do when attacked.
In the situation you describe the bandit has both a position and e advantage. The way to survive and win is to neutralize these advantages starting with position since position is what allows him to shoot you, not e. Baldeagle's suggested maneuver is certainly valid but is dependant upon how much e and altitude each of you have along with how smart the bandit is. If you're slow (say you're jumped on climb out) he may catch you before you gain enough e to pull the g necessary for him to overshoot. Also, if he's smart he'll throttle back, lag your turn a bit and then saddle up on you.
On the other hand, if you turn into him you'll negate some, maybe all, of his position advantage even though he maintains his e advantage. If you can pass close aboard and a full 180 out (beak to beak) you've neutralized his position advantage and you can then immediately extend for separation while he has to complete a 180 degree reversal to get his nose back on. Will you gain enough separation to get away? It depends on your comparative aircraft performance and altitude but probably not; however, by forcing him to turn 180 and then chase you, you're equalizing e-states by making him drive a longer horizontal distance. When you see him getting his nose back in your direction be prepared turn into him again and repeat the process until you've neutralized his e advantage.
This is certainly not the only way you can handle this but there are the fundamental ACM rules such as turning into your opponent and then there are the "graduate level" tactics. Since it sounds like you may be relatively new, work on negating the threat first and then work your way up to the higher level (and usually risker) tactics. Let me add one more thing. Will turning into the threat mean you'll "win". No, of course not. Anytime a bandit has an advantage at the start of the fight it's a very very tough situation for you and you'll lose more often than you'll win or escape but you will survive longer and learn while doing it.