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Topic: Interesting Probability Problem for Serious Geeks: Self-correcting
Random Walks

Replies: 0

 vincent64@yahoo.com Posts: 129 Registered: 4/18/07
Interesting Probability Problem for Serious Geeks: Self-correcting
Random Walks

Posted: Oct 4, 2017 7:09 PM

This is another off-the-beaten-path problem, one that you won't find in textbooks. You can solve it using data science methods (my approach) but the mathematician with some spare time could find an elegant solution. Share it with your colleagues to see how math-savvy they are, or with your students. I was able to make substantial progress in 1-2 hours of work using Excel alone, thought I haven't found a final solution yet (maybe you will.) My Excel spreadsheet with all computations is accessible from this article. You don't need a deep statistical background to quickly discover some fun and interesting results playing with this stuff. Computer scientists, software engineers, quants, BI and analytic professionals from beginners to veterans, will also be able to enjoy it!
The problem

We are dealing with a stochastic process barely more complicated than a random walk. Random walks are also called drunken walks, as they represent the path of a drunken guy moving left and right seemingly randomly, and getting lost over time. Here the process is a self-correcting random walk, also called controlled random walk, in the sense that the walker, less drunk than in a random walk, is able to correct any departure from a straight path, more and more over time, by either slightly over- or under-correcting at each step. One of the model parameter (the positive parameter a) represents how drunk the walker is, with a = 0 being the worst. Unless a = 0, the amplitude of the corrections decreases over time to the point that eventually (after many steps) the walker walks almost straight and arrives at his destination. This model represents many physical processes, for instance the behavior of a stock market somewhat controlled by a government to avoid bubbles and implosions, and it is defined as follows:
Let's start with X(1) = 0, and define X(k) recursively as follows, for k > 1: [...]