In this seminar, Ihor Kendiukhov presents the Ergodicity Library – a free, open source, and comprehensive Python toolkit developed specifically for research in ergodicity economics and related fields. The library features tools for simulating continuous stochastic processes, training artificial agents to operate under uncertainty, and advanced visualization. It places emphasis on fat-tailed and multiplicative processes, especially Lévy stable processes, given their importance in economics.
The seminar was hosted by Emilie Rosenlund Soysal (London Mathematical Laboratory) and James King (Science Practise).
Congratulations Ihor: very well done, a great tool, I am going to use it!
A general question for the broader audience.
I am studying ergodicity economics with the eventual goal -beside the fun of learning such a novel and stimulating subject- of integrating the perspective it provides with fat tails-based statistics.
Where do we stand concerning the connection of non-ergodic dynamics with fat tails statistics? I get it for Lévy stable processes, alright. Any review of the state of the heart available out there?
Thanks you in advance.
Hi Mirko, great question!
As far as I know, there are currently no publications explicitly applying the EE approach to processes with fat tails. However, I am currently working on a blog post that touches on this topic. Fat tails frequently come up in internal discussions at LML, so based on what we have talked about, let me briefly tell you my ideas on the issue (illustrated with the Levy-stable processes).
To apply the approach to decision making outlined in the EE textbook, three questions need to be asked (in order): For a given process,
(1) does its ergodicity transformation exist?
(2) if yes, does the time average growth rate converge to a scalar (almost surely)?
(3) if yes, how long do you have to wait?
For the GBM and other geometric Levy-stable processes, the answer to (1) is yes (log transformation). For geometric Levy-stable processes with alpha > 1 (including GBM with alpha = 2), the answer to (2) is also yes – indeed, the time average growth rate converges because the noise of the finite-time average scales with t^(1/alpha-1), which for alpha>1 means the noise declines with an increasing t. We can then move on to question (3): Again, because the noise of the finite-time average growth rate scales with t^(1/alpha-1), smaller alpha means slower convergence. In conclusion, as long as the tails are not extremely fat, the “only” impact of the tail shape is how long one has to wait for the EE decision criterion to become meaningful.
In the case of extremely fat tails (alpha ≤ 1), the answer to question (2) is no: the distribution of the finite-time average growth rate does not collapse to a scalar in the long run but is instead given by a full distribution. Hence, when comparing such processes, it is impossible to say with certainty what is optimal in the long run—regardless of how long you may wait.
I would personally love to see a similar analysis of many other processes with (or without) fat tails. I suspect that some processes would fail already at the first question, (e.g. all the processes with ruin as an absorbing state), while for others it is really a matter of convergence and convergence time. If this is something you are thinking about looking into, please keep in touch!
//Emilie
Hello Emilie,
apologies for my late reply.
First of all, thank you for your thoughtful answer.
Second, please let me know where you will post this detailed analysis. I would like to join my evolving understanding of these two fields as much as possible and this is all gold in this respect.
What you are saying goes in the direction of confirming what I suspected but could not properly articulate. I will just rephrase it to make sure I understand it the hand-waving way from my perspective.
If one knew a lot about statistical distributions but nothing except for the main idea of EE (distinction between average across samples and average along time), for GBM these three convergences could be -at least- intuitively guessed from its very definition, because the defining stochastic differential of GBM includes \sigma as a pre-given, finite variable. This is a priori a very strong constraint on the ensuing distribution.
And I also understand intuitively that, when the resulting distribution does not have a mean, as it happens for the wildest financial assets, then it is to be expected that the events that really make a difference can take an extremely long time to happen, so the multiplicative growth rate is going to be affected massively.
I don’t know if it makes sense.
Looking forward to reading your blog post.
All the best,
Mirko