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

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.

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.

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.