Random Walks, Brownian Motions, and Related Stochastic Processes

Entitled “Random Walks, Brownian Motions, and Related Stochastic Processes”, the full version in PDF format is accessible in the “Free Books and Articles” section, here. This article is an extract from my book “Gentle Introduction to Chaotic Dynamical Systems”, available here.

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.

Top: Brownian (green), integrated (orange), Bottom: reflective random walk

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.

Table of Contents

Download the Article

The technical article, entitled Random Walks, Brownian Motions, and Related Stochastic Processes, is accessible in the “Free Books and Articles” section, here. It contains links to my GitHub files, to easily copy and paste the code. The text highlighted in orange in this PDF document are keywords that will be incorporated in the index, when I aggregate all my related articles into books about machine learning, visualization and Python, similar to these ones. The text highlighted in blue corresponds to external clickable links, mostly references. And red is used for internal links, pointing to a section, bibliography entry, equation, and so on.

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About the Author

Vincent Granville is a pioneering data scientist and machine learning expert, co-founder of Data Science Central (acquired by  TechTarget in 2020), founder of MLTechniques.com, former VC-funded executive, author and patent owner. Vincent’s past corporate experience includes Visa, Wells Fargo, eBay, NBC, Microsoft, and CNET. Vincent is also a former post-doc at Cambridge University, and the National Institute of Statistical Sciences (NISS).  

Vincent published in Journal of Number TheoryJournal of the Royal Statistical Society (Series B), and IEEE Transactions on Pattern Analysis and Machine Intelligence. He is also the author of multiple books, including “Intuitive Machine Learning and Explainable AI”, available here. He lives  in Washington state, and enjoys doing research on spatial stochastic processes, chaotic dynamical systems, experimental math and probabilistic number theory.

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