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

Question

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 1

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.

Answer 2

This question depends on many things. Not only does it depend on the country, but it depends on the level of university you plan to attend. I will focus here on the US system. Various programs include the masters program as one moves on to obtain a PhD; however, many universities will expect, as you said "Complex Analysis, Real Analysis, Linear Algebra, Abstract Algebra" and possibly some geometry/topology courses.

Many programs expect a core amount of knowledge of these subjects. If you are looking at places that are typically regarded as top 20 or so, then you will probably run into difficulty in admissions. Otherwise, if you look at a program's website, they will usually tell you the classes that they require.

If you are coming from a small school, then sometimes programs will make an exception if they really like your file and give you extra time to catch up in a program, as you did not have the opportunity to take the required courses and give you a special deal. This is rare.

Answer 3

My experience (in Comp Sci, but on the mathematical side; and doing some engineering work in the 'Hi-Tech' sector while my research was pure theory) suggests the following:

• Don't worry about the titles (Master, Doctor etc.) but about what you're trying to accomplish, personally and research-wise.
• Don't consider working on doctoral-scale research unless you have a specific subject you're interested in studying (much more specific than stochastic analysis in general). Don't start a 10-mile race when you're not sure where you intend to get to. Yes, I know that some people start graduate programs without a clear research subject, and it's not impossible, but I don't recommend it.
• If you don't have a subject, I'd suggest finding other employment, in or out of Academia, and studying a bit on your own, maybe taking a course or two here-and-there, to see if something more specific piques your interest. An alternative to that is doing a Master's (but this kind of depends on the country you're in, like other answers suggest), and sort of dipping your feet in the water. Note: In some countries / academic cultures, Masters' programs are not well-regarded and you're expected to not go through them before doing a Ph.D. (e.g. in the US); in these cases, and if you're into pure math, you might not get reasonable inspiration from working in industry.
• Stochastic analysis is hella difficult, or at least that's how I felt when I learned some fundamentals of Ito Calculus. I'd build myself up to it a bit...
• If you have a subject, find an advisor. Yes, before beginning. Even if someone doesn't agree automatically, immediately, or at all to be your advisor - they might still give you some solid advice, based on more specific information about your background, regarding which courses you might want to take, books you might need to read, and experience you might need to gather before you're sorta-kinda-ready. Or they might very well say "you've got 4/5/6 years, use the first 1-2 years for catching up." This bring me back to the first point: If you're in synch with an advisor, let him/her worry about arranging the formalities of the process you'll undergo; or at least work out some sort of speculative plan with him/her.