Hi, I will probably get into a thesis-based master program in financial engineering this fall. With the thesis option, it will be very flexible for me to select courses and research topic. As I become more and more interested in mathematics, especially analysis, I am considering applying for a math PhD afterwards. So I plan to select more graduate level pure math courses in addition to my two required finance courses. The courses that I want to take are harmonic analysis, functional analysis, probability theory, stochastic calculus, and operator algebra. I think I can handle these demanding courses because I took some advanced math courses for my bachelor's degree, such as real analysis, measure theory, integration theory, time series, applied probabilities, ect..
My questions are -
1. Will it be realistic, or does it look odd that I want to apply for a math PhD following a financial engineering master program?
2. Will the financial engineering program director allow me to take those pure math courses? (I can choose a demanding thesis topic that requires a lot of pure math knowledge. In that way, I guess I can convince them to allow me to take those courses.)
3. What are my chances of getting into a top 10 math PhD program, like MIT, Princeton, Stanford, Berkeley? (FYI, I have a good undergraduate GPA, 91%-94%, with a major in math/finance, at a quite famous university. For my master thesis, I will probably work on stochastic analysis which will involve a lot of pure math knowledge, and I will try to get around 95% GPA for my master study.)
4. I am thinking of working with a finance professor and try to write some papers, which will involve a lot of math stuffs. Will that be helpful in proving my research ability or just a waste of time considering those are not pure math stuffs?
Thanks in advance for your advice!