graduate admissions - With a background in mathematical finance and desire to apply for a mathematics PhD in another field, do I need a second master's?

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. 
   

   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. 
   

   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. 
   
   

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

   
   


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.

   
   


5. 
   

   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.

   
   


6. 
   

   Furthermore, while I technically passed the thesis, I scored the lowest passing mark. Meanwhile, I did score well in all the classes relevant to stochastic analysis.

   
   
   

Answer

I would say the answer depends on your background in mathematics, and from the information you give, I can't tell how extensive that is. It isn't so important what the title is on your degree; it matters what you've done and what you know.

To start a US-style PhD in pure math, in which the first couple of years are coursework, you should have the equivalent preparation of a bachelor's degree in mathematics. It should include, for example, about one year each of proof-based real analysis and abstract algebra. You should be very comfortable with reading and writing mathematical proofs.

To start a European-style PhD, you are expected to be ready to begin working on research. You should have completed graduate-level courses in algebra and analysis, and be very familiar with fundamental material in your subfield. For an area like stochastic analysis, for example, I'd say you'd want to know the majority of the material in a textbook like Karatzas and Shreve, and be able to work through the proofs and solve the exercises.

One question is - what was the level of mathematical rigor in your courses up to stochastic calculus? If it wasn't high, that's likely to be a problem.