Does this graph look like a kid’s drawing? Maybe a piece of art from the monkey Jeff? No, of course Jeff draws better than this.

Actually, it’s a representation of what is known as a Lévy flight; a mathematical concept that shows up in nature, marketing, cryptography, astronomy, biology, physics and nowadays even in social media.

Here are a couple of examples of Lévy flights for “dummies”:

- A shark that forages will remain in a small area, looking for fish. Then it will realise it has used up all the sources in this spot. It will then head off in a random direction, travel some distance, and start foraging again.

- Someone discovers a website. He visits it once, and then returns and returns again. Then he feels that there’s no more benefit and he moves on, surfing until he finds another website to visit.

So… a Lévy flight seems to be a set of **random short movements** connected by infrequent longer ones.

The formal definition for this concept is “a** random walk** in which the step-lengths have a probability distribution that is heavy-tailed”. We can say that it is a random walk with the particularity of showing heavy jumps because the step lengths of this process come from a distribution with infinite variance.

As with any stochastic process, Lévy flights have their origin in diffusion processes. Because of this, they are useful in stochastic measurements and simulations of random or pseudo-random natural phenomena. Particularly, they show an anomalous diffusion: there exists a kind of “microstructure” in the system and due to this fact, it is related to** chaos theory**.

As we have already mentioned, this field has many applications explaining several types of behaviour present in nature. Recent biological research has shown that the combination of Lévy flights with Brownian Motion describe patterns in animal’s hunting. But what are the main differences between both processes?

## Differences between Lévy flights and Brownian Motion

A Lévy flight belongs to the family of alpha-stable random variables, as does a Gaussian Process (and also a Brownian Motion, as the one showed in the picture on the left).

However there are also key **differences between both processes:**

Although the most extended model for stock prices is Brownian Motion, many people think that an alpha stable distribution would be more accurate. This would consist of viewing the price of a stock as an accumulation of jumps assumed to be approximately i.i.d. from a heavy-tailed distribution (with infinite variance).

Let’s look at these two variables, the price of a Spanish stock and the price of the stock index to which it belongs.

We could say that this is more similar to a Lévy flight than to Brownian Motion. And, finally, this leads us to the next question:

*Are we able to find any advantages from Lévy flights in order to improve activities related with the stock markets?*

We will investigate it in the future…

See you next time!

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Interesting post!

There is another way to detect if a stock follows a Pareto-Lévy distribution. Plot the returns of the stock, then see if they deviate substantially from 0 within one or two standard deviations. Plot the theoretical returns according to a brownian motion (normal distribution) and a levy-flight (power-law). You’ll see that rare or extreme event are better adapted with the second. All this is very well explained in Mandelbrot’s book “The (mis)behaviour of markets” that you most probably know.

Thanks for the suggestion Alvaro! I will take it into account!

Thank you for sharing this amazing information