I'm a Scandinavian student currently in a 5 year MSc in industrial engineering program (you go into it directly from high school, first 3 years = bachelor, 2 last years = master). I'm in my third year now and I've started thinking about pursuing graduate school. Unfortunately it is a little to late to switch major, but I'm planning on taking as many math classes I can and my masters concentration will be in financial engineering. Starting in the fall I will do a year abroad at a top 10 US engineering school. Currently I've taken the following math courses: calculus in one variable, calculus in several variables, linear algebra, mathematical statistics basic course, analytic functions, systems and transforms, optimization (will take stochastic processes and numerical analysis before summer). Also of course engineering courses and a few finance courses. My grades so far are 4,6/5,0 with top grade in all but one math course.
Okay given my background, what courses should I take to prepare me for graduate studies? I've read a lot about real analysis being necessary. As part of my concentration in financial engineering I will plan to take financial statistics, Monte Carlo and empirical methods for statistical inference, statistical modelling of extreme values, econometrics and a few more. What I'm worried about is that my math background isn't as rigorous as someone who majored in math. For example I have no discrete mathematics, topology etc. Will this be a big problem? Should I apply directly to a PhD or to a MS in statistics/math first? Also my university has a master in math statistics which I can take and count some of my credits from my MSc in engineering. Though it would mean that my graduation would be prolonged by a year.
Would really appreciate some insight on the topic