Wednesday, 19 February 2014

Lecture 15

The Wikipedia page about random walk is worth a look. Today, we focused upon the Gambler's ruin problem. However, there are many more interesting things that can be said about random walks. For example, one can consider random walks on arbitrary graphs. More about random walks will be presented to you in the course Markov Chains, Part IB.

I mentioned that if you wish to ensure you never go bankrupt then rather than bet £1 at each go you could bet £ $fz$, where $0<f<1$. That way you come next to $(1+f)z$ or $(1-f)z$. Kelly betting, that I will talk about later (probably in Lecture 21), is about choosing the optimal size of $f$.

The continuous version of random walk is Brownian motion, which is presented in Stochastic Financial Models, Part II. One of its beautiful properties is that it is self-similar. The graphic below shows how Brownian motion looks when we place it under the microscope and gradually increase the magnification. It is possible to say of Brownian motion in 2 dimensions that "almost surely every arc of the path contains the complete works of Shakespeare in the handwriting of Bacon." (Peter Whittle). Just as we might say in dimension 1 that at some magnifications the path below looks like a graph of FTSE all-share index for the past 5 years.