What is the typical PhD application procedure?

Part I Background on Myself

Part II What I Heard

Part I

I am nearly finished with my master in mathematical finance. I have finished my thesis and have one course left to take.

I am interested in taking up a PhD in stochastic analysis.

Part II

1 Read up various literature on your desired dissertation topic which should include several textbooks and even more for recently published articles.

2 Come up with a PhD proposal.

3 Contact relevant faculty of the universities to which you intend to apply.

4 Discuss your proposal with them if they're up to it.

5 One of the ff: Revise proposal if needed, completely change proposal or get referred to different faculty member.

6 Actually apply to the university.

So, it seems that PhDs unlike applications for master's or bachelor's involve speaking first to faculty and then to the school? That is, when I apply I should already have a professor in the university already to take me on as a doctoral advisor?