It is possible for you (on your own) to file your own motion to terminate your probation early under California Penal Code 1203.4 and related provisions. There's a good explanation of the process at http://www.saclaw.org/Uploads/files/Step-by-Step/ExpungingCriminalRecords.pdf . You can file your own motion under to terminate probation early, to reduce your felony to a misdemeanor and have the conviction dismissed. Those are things you can ask the court to do and the forms are pretty easy to find online.
However, there are categories of convictions that will not be eligible for this process. You should check the website above as it has lists of ineligibile conditions.
Trying to get your probation officer on your side (if you can get ahold of them) is also helpful. If you've been a "model probationer" Probation may be able to help grant you certain exemptions or changes.
Most court's won't even seriously consider a motion to terminate probation early until you have covered more than half the time. They definitely will not until you have paid all fines and fees. If you meet these criteria and have a good reason (like applying for a job that you could get if you weren't on probation) you probably have a good chance. You also need to not be serving any sentence and not be facing charges now in any other case.
NOTE: If you make this motion too early there may be problems. By law you may only make this motion once, however, most judges, if you or your attorney ask, will delay the motion for sufficient time to allow you to do what they want before they grant the motion.
To have even the slightest chance of success, you have to have paid all your fines, fees, and other charges. You also to have completed all other conditions of probation. Even then, early termination is a hard sell.