The good thing, is that you can fix the problem by sending it to repair...There are lots of places you can check that do this service. Replacing the analog stick might not be something you dare do yourself, because it might be too complicated to open up the PSP to do the installation yourself. Although I know many gamers that are very techy that way... but not all though and this also includes me. I would rather send it to repair like I once did when my PSP got bricked.
Now there might be places around where you live that do PSP repairs, and I would suggest THAT is the first thing to look for. Why because you are prompt to have results and you are dealing with the one who is going to do the job for you face to face, instead of sending it somewhere you searched on the internet, and might take days or weeks before you get good results. If no, then check places on the internet, as second option...maybe do a Google search typing "PSP repairs" or "PSP analog stick repair". or something the like. I am sure you are going to find good results. But your work does not stop there, make sure to do some type research on the company you are dealing with and find out how they will solve your broken PPS analog stick, or other problem.
If you are just looking for the PSP analog stick replacement you can Find It Here For Really Cheap
I hope this has been helpful! have a nice day, and thank you so much for the visits :D