Pardon the dumb question if ever. I notice similar questions all sound like "can I get a second master's?" but mine is "do I need?"

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

Edit: I am about to graduate from a third world university and am planning to apply for a PhD in a first world university.

I am interested in taking up a PhD in a different branch of mathematics called stochastic analysis (stochastic calculus. or whatever it's called.) but am wondering if my background is insufficient. If so, I may take up a second master's.

Given my limited background in stochastic analysis and other information (below), can I already apply for a PhD with stochastic analysis for my dissertation topic, or might I need a second master's first?

Some information:

1/4 I am not particularly interested in mathematical finance anymore unless it is in a rigorous context rather than with computers, modeling, statistics, non-mathematical finance, simulations, etc. As of right now, I no longer have any plans to go into industry.

2/4 My background in Stochastic Analysis is 2 courses on Stochastic Calculus, 1 prerequisite course for Stochastic Calculus (probability) and 2 halves of classes which apply the stochastic calculus.

3/4 I don't know about the basics of one kind of stochastic calculus as we were taught mainly the other kind.

4/4 My thesis is about credit risk, having mainly to do with statistics, the part of mathematical finance I don't want to go in to, if I were still interested in mathematical finance. Hence, I feel have no background in mathematical research given that the research conducted was mainly to do with statistics and finance.

I have read a sample PhD proposal and have no clue what to put for methodology. I am not quite sure what kind of research methods pure mathematicians have.

Furthermore, while I technically passed the thesis, I scored the lowest passing mark.