My initial reply was just messin'. I've answered this question so many times it's..........
You say that he's much faster so don't chase him around the ring. Be patient and make him come to you.
Try this. Instead of playing the pawing game with his jab fist and yours let him throw his jabs at you. Brush the first few to the side until you get his rythme and timing. When you think you got it just slip his jab to the left and put your right hand out, he'll run right into it. Dig a hook to the body and take the side. From there you are in absolute control.
Also, lead rights to the body. Again, let him come to you. When you see that he is attacking just get down low with your legs and send that right hand straight into his heart. Ofcourse follow with hooks take the side and again you are a step ahead.
Try feinting him aswell. Feint him, look for how he reacts and work off of the knowledge when you do it the next time.
Start with those and let me know how it goes, there are many more ways to skin a southpaw............Rockin'