# What mathematical background is needed to study random walks rigorously?

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?

• Real Analysis because you need it for probability. But if you are doing this for purely finance, I think PDE (which eventually requires you to study analysis...) – IAmNoOne Aug 3 '16 at 22:34
• It almost sounds to me like what you know is enough to study random walks rigoroulsly, for applications in finance. @Nameless Why is Real Analysis is necessary for a study of random walks in finance? Just solid Riemannian calculus should be enough no? I don't think he needs measure theory to study random walks in finance. Or are we taking about things like brownian motion? Also I don't think you need PDE's. – Gregory Grant Aug 3 '16 at 23:01
• Try reading a rigorous book on random walks and when you come to something you don't know, go study up on that in particular. I think that's a better approach than trying to learn measure theory and PDE's first. – Gregory Grant Aug 3 '16 at 23:01
• Pick up a book on Markov Chains and/or Martingales, basically anything covered in a first semester grad course in probability.The simple symmetric random walk in $\Bbb Z$ or $\Bbb Z^d$ is a special case of these more abstract objects, and learning the more abstract theory will help you appreciate what's behind them in more detail. I personally like these notes by Friedli covering Markov Chains: mat.ufmg.br/~sacha/Textos_Diversos/Cadeias_Markov.pdf – Shalop Aug 3 '16 at 23:37
• If you just want a self-contained introductory book without any abstract measure-theoretic constructs, try this book: math.dartmouth.edu/~doyle/docs/walks/walks.pdf – Shalop Aug 4 '16 at 0:06