A process for preparing copolymers which comprises copolymerizing (A) at least one ethylenically unsaturated dicarboxylic anhydride derived from at least one dicarboxylic acid having 4 to 8 carbon atoms, (B) at least one vinylaromatic compound, and (C) optionally at least one non-(A) ethylenically unsaturated comonomer containing at least one heteroatom, (D) reacting the copolymer with at least one compound of the formula I a or I b where A1 is C2-C20 alkylene, identical or different, R1 is C1-C30 alkyl, linear or branched, and n is an integer from 1 to 200, hydrolyzing the product with water or an aqueous alkaline solution, and subsequent to hydrolysis adding further free-radical initiator.