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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?

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    $\begingroup$ 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...) $\endgroup$ – IAmNoOne Aug 3 '16 at 22:34
  • $\begingroup$ 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. $\endgroup$ – Gregory Grant Aug 3 '16 at 23:01
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    $\begingroup$ 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. $\endgroup$ – Gregory Grant Aug 3 '16 at 23:01
  • $\begingroup$ 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 $\endgroup$ – Shalop Aug 3 '16 at 23:37
  • $\begingroup$ 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 $\endgroup$ – Shalop Aug 4 '16 at 0:06
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I'm a little less optimistic than the other respondents as regards the required background to study mathematical finance. I've seen a lot of students struggle in upper level courses due to weak backgrounds in analysis.

Of course, if you just want to stick to discrete time models, then you can get away with less background. I'd recommend Shreve's Stochastic Calculus for Finance I: The Binomial Asset Pricing Model if you are going to take this route.

But if you serious about doing continuous time finance, you need to understand Brownian motion, quadratic variation, sigma algebras, etc. which means a good understanding of real analysis up through Lebesgue integration, and preferably a measure-theoretic introduction to probability. If you try to do SDE's without that, it is just formal manipulation of expressions, without really understanding what is going on mathematically.

That said, a deep mathematical understanding is not per se necessary to succeed in the financial industry, so it depends on what your goals are.

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You seem to know most of what is required. I would get comfortable with the Central Limit Theorem, if you aren't already.

Here is a source on Random Walks you might start with: http://www.mit.edu/~kardar/teaching/projects/chemotaxis%28AndreaSchmidt%29/random.htm

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