If the bullying is violent then technically its assualt. Therefore of course they shouldbe allowed to prosecute. Though i think it already is possible to prosecute for bullying. If a child is beaten by another child, that child can face criminal charges just as if anyone else beats them. Also, whether it occurs in the school grounds or not is irrelevant. There is nowhere you can hide from the law! Obviously you onlyreally want as a last resort but i think its possible already.
First, someone who bully others, having something wrong during his/her lifetime. This action may attribute to parent's education, culture, or school. But the people who suffered from a bully needs their right back.
Second, maybe making a prosecution is too extreme, however, if the victims couldn't do anything, the bully wouldn't stop.
Therefore, we can not only make use of education to decrease the rate of bullying but also ask the bully for what they have done.
Teachers have a responsibility to protect their students. If they took that responsibility seriously ans stayed aware, the bullying would never have gotten to the point of prosecution. Prosecute the school, not the bully. The bully is a child who cant make good decisions, and jail or fines will not internalize anti bullying messages.