Introduction to Random Walks, Brownian Motions, and Related Stochastic Processes
In about 15 pages, this scratch course covers a lot more material than expected in such a short presentation. It constitutes the first chapter of my upcoming book “Gentle Introduction to Chaotic Dynamical Systems”. Other books in this series are available here.
Written in simple English yet covering topics ranging from introductory to advanced, it is aimed at practitioners interested in a quick, compact, easy-to-read summary on the subject. Students learning quantitative finance, physics or machine learning will also benefit from this material. It is designed to help them understand concepts that are traditionally embedded in jargon and arcane theories.
There is no reference to measure theory: the approach to even the most advanced concepts is very intuitive, to the point that it is suited to high school students taking advanced classes. Most of the material deals with stochastic processes less basic than the standard Brownian motion and random walk. In particular, I discuss integrated and doubly integrated Brownian motion, and 2D Brownian-like processes exhibiting a strong clustering structure. Reflective random walks lead to the concept of invariant measure (the limiting distribution of the process) obtained by solving a stochastic integral equation. I show how to do it numerically in Python. In this case, the exact solution is known, and can be compared with results obtained via simulations. I also discuss constrained random walks, and the Hurst exponent to measure the smoothness of such processes, with illustrations. While technically, the derivative of Brownian-like processes does not exist, I show how you can make sense of it: it leads to interesting shapes (not math functions) with a fractal dimension. A lot of emphasis is on creating a rich class of processes, each with specific features. The goal is to show how to generate them, and in which ways they are distinct from each other, in order to use them in applications.
Read more, download the article and access the Python code, from here.
