academic career in stochastic processes

Question

I am second year bachelor math student. I made a decision to devote myself to the study of probability and stochastic processes. Right now I know calculus, little bit of analysis (Abbott), some linear algebra(Axler), some probability and math statistics.

My question is what topics should I learn to prepare myself for the study of stochastic processes at the very high level? What courses will develop my general understanding of the stochastic processes?

What courses should I take in parallel with probability courses? Shall I learn functional analysis or go deeper into real analysis and study Rudin's book etc?

I will be very grateful to hear some general advices for studying stochastic processes. If this site isn't appropriate place to ask such questions, I am sorry.

Answer 1

First of all, don't be sorry :) This is definitely a good place to ask such a question. Maybe more experienced users can help you tag this question better.

In my opinion, your time will be well spent right now if you focus on making your foundations in calculus and linear algebra strong. Once you're at the level where you have a decent understanding of Rudin, I would suggest learning Measure Theory as that will enable you to access most of the texts in Probability theory and Stochastic processes.

The part of stochastic process I find most exciting is the study of systems with interacting particles (see reference: http://math.arizona.edu/~sethuram/588/Liggett.pdf). Of course, a good understanding of non-interacting particle systems is recommended before you move on to this. However, that can be achieved through most standard courses in stochastic processes.

I'll be happy to answer to more specific questions, if you have any.

Answer 2

While I'm happy to make a few suggestions for courses that support an interest in stochastic processes, if I were your advisor I'd also stress that your undergraduate years are ones for you to explore and possibly find additional areas of interest. Note that most people change careers five or six times during a lifetime, so don't become too narrow too soon!

Anyway, it is essential you become a good programmer, so take a course (or learn on your own) and use $R$, $C++$, $Matlab$, $Mathematica$ or other powerful programming language. Next: Information theory, which is always a great foundation, especially if you'll be working alongside engineers. Take advance statistics—not for data analysis, but for stochastic processes. Although not essential, you'd profit from queuing theory. Of course statistical mechanics in Physics is relevant, but I suspect a Physics department will demand you take three or four physics classes before you get to that, so it is probably better to approach this through a Math or Statistics or Operations Research department.

Good luck!