I've been studying random walks as part of a finance course and I'm starting to become really interested, and I'd like to start a study of them from a mathematics (i.e. more formal/rigorous) point of view.
However, my knowledge of mathematics is quite limited (calculus, basic linear algebra and probability, a bit of (stochastic) differential equations). So I was wondering what fields I need to familiarise myself with before I can effectively study random walks. Maybe there's even a good self-contained book on random walks for beginners?