If you're on the tourist trail in Roma, you'll find tourist/Roma prices. If you go out of the city, you might find different stores and different pricing. If it's only for 4 months then go cheap and cheerful - iPer, for example, sells clothing.

In the case that you're prepared to compromise and not have the jacket that you have in your mind's eye, you'll find lots of options out there. You might want to get one you can ski in too, if you're planning that.

You're a student, you have no street cred

Layers are also good and are flexible