graduate admissions - Does one need a master's in math before taking a PhD in pure math?

I finished bachelor's in mathematical finance and am nearly finished with master's in mathematical finance (I am already done with thesis), and I plan to pursue a PhD not in mathematical finance but in pure mathematics particularly stochastic analysis.

Is getting into a PhD in pure mathematics possible without a master's in pure mathematics? If so, how can I best prepare for it what difficulties may I encounter?

I have two concerns in particular:

1. I feel I do not have enough training in mathematical research. In undergraduate studies, we did not have many mathematical research projects. Some of our projects included researching on particular topics involving applications of mathematics we learned (since we were an applied mathematics course) and problem sets, but I don't know for how much they count towards mathematical research experience. We had some statistics and finance projects, but obviously they don't count.

We did not have a thesis in undergraduate studies, and most of our theses in master's did not involve much pure mathematics (which in mathematical finance would be stochastic analysis since as far as I know no other non-statistical math is used in mathematical finance). I have a hunch none of us this batch or in the batches before us ever had to research in mathematics for our/their theses.

2. I do not have much exposure to other kinds of mathematics. One of my coursemates helped me realize that one of my reasons of choosing stochastic analysis is our limited exposure to other math. I was aware of this but did not think this was a problem.

As far as I know, MS Math programs require Complex Analysis, Real Analysis, Linear Algebra, Abstract Algebra and then some electives and thesis. I don't think the lack of classes is a problem as I guess I can take those during the PhD program. To me, it seems my concern is the lack of a mathematical thesis.

So, is my limited exposure to other math a problem?

Our math classes besides Calculus I, II, III, Linear Algebra and Elementary Probability are:

1. 
   

   1 class of each: ODE, PDE, Discrete Mathematics, Numerical Analysis/Scientific Computing, Elementary Real Analysis (the one with Riemann-Stieltjes), Advanced Real Analysis (the one with Lebesgue), Advanced Probability (the one with Measure Theory)

   
   


2. 
   

   4 Statistics classes. (As I like to put it, "More statistics than I'll ever use in my life.")

   
   


3. 
   
   

   NO Complex Analysis, Abstract Algebra, Topology, Graph Theory or Number Theory (though the last 2 are in our discrete mathematics, they weren't taught in our discrete mathematics classes).

   
   


4. 
   

   2 Stochastic Calculus classes

   
   

This comment says I should be "be comfortable with mathematical proof in a variety of areas"

Answer

It depends on the country.

In the US, it is frequent that students go from undergrad right into their PhD. While there, some do get a Master's, but as a side effect of coursework for the Ph.D. Though, I myself did get a Master's first.

I believe it is more common in Europe to get a Master's first due to the shortened Ph.D. process there.