Jurassic


Total Posts: 290 
Joined: Mar 2018 


What does it mean for financial markets to be ergodic or nonergodic processes? Can anyone explain this without having to go into the mathematics too deeply. 




Maggette


Total Posts: 1187 
Joined: Jun 2007 


I think the wiki definition is as precise as you can get for a verbal definition:
"In econometrics and signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long, random sample of the process. The reasoning is that any collection of random samples from a process must represent the average statistical properties of the entire process. In other words, regardless of what the individual samples are, a birdseye view of the collection of samples must represent the whole process. Conversely, a process that is not ergodic is a process that changes erratically at an inconsistent" 
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ahd


Total Posts: 29 
Joined: May 2017 


Roughly speaking, a process is ergodic if the rules that govern its time evolution are 1) constant and 2) iid over some differencing timescale.
There's no "time" in statistics. Statistics, the central limit theorem, etc. are derived to tell us something about the properties of a big ensemble of experiments run in parallel. But the world evolves in series, moment to moment. The assumption of ergodicity allows you to use methods that are derived to be valid on a set of measurements made in parallel to a set of measurements made in series. 




deeds


Total Posts: 462 
Joined: Dec 2008 


Can the markets be both reflexive (Sorossense) and ergodic? 



Maggette


Total Posts: 1187 
Joined: Jun 2007 


That IMHO depends how the feedback manifests itself. 
Ich kam hierher und sah dich und deine Leute lächeln,
und sagte mir: Maggette, scheiss auf den small talk,
lass lieber deine Fäuste sprechen...





Heuristically, reflexivity implies some sort of path dependence which would contradict ergodicity. Developing the formalism to show this rigorously may not be worth the pain (I’ve never seen a mathematical formulation of the reflexivity theory, but to be fair, I haven’t really looked either)...




ronin


Total Posts: 527 
Joined: May 2006 


Sure it can.
d Price = (Sentiment  Price) dt / T + sigma dW d Sentiment = (Price  Sentiment) dt / S + eta dZ
The pair (Price, Sentiment) is ergodic and bivariate normal. But Price alone isn't.
The introduction of Sentiment generates forward arbitrage, so it isn't consistent with finance as we know it.

"There is a SIX am?"  Arthur 



Jurassic


Total Posts: 290 
Joined: Mar 2018 


ok thanks guy.
On a somewhat related topic, how does this differ with respect to stationary processes in able to apply statistical techniques (or the like)? 



day1pnl


Total Posts: 54 
Joined: Jun 2017 


Roughly speaking a process is stationary if the shift operator is a measure preserving mapping on the space of sample functions of the process, i.e. the image measure of the distribution of the process under the shift operatoris equal to the original distribution of the process itself. Such stationary stochastic process is then called ergodic if additionally the shift operator is ergodic on the space of sample functions (i.e. the invariant sigmaalgebra for the shift operator is trivial so that all sets have either prob = 1 or 0).
So basically ergodic implies stationary by definition.
The point about the invariant sigmaalgebra comes here: Empty set Ø and the universe Omega share that they are both invariant under preimages: Preimage T^(1)(Ø)=Ø and preimage of T^(1)(Omega)=Omega. The invariant sigmaalgebra simply consists of all sets F that have this property T^1(F)=F. The ergodicity requirement requires that sets that "who want to behave like the universe or empty set" also have probability either 1 or 0 (like the universe or empty set).
If you think about what it means that a set is preimeage invariant for a mapping (for example shift operator) it basically means that the transformation "gets stuck" in a particular small part of the universe: the ergodicity ensures that does not happen.
Re: What does it mean for financial markets to be ergodic or nonergodic processes?
I guess it means intuitively that if financial markets are ergodic that the market will over time go everywhere and its historical distribution is sort of bound to repeat itself over and over again... 





Has anyone been following Ole Peters? Ergodicity Economics?
His complete lecture notes (136 pages): ergodicityeconomics.files.wordpress.com/2018/06/ergodicity_economics.pdf
His basic thesis is that mainstream economics has somehow confused timeaverages with ensemble averages, with ergodicity being the necessary condition for treating the two interchangeably.
I wish he would be more direct and just say don't forget the Ito drift when dealing with timeaverages. The ergodicity argument seems to be more general and widelyapplicable in his view.
One of his papers was discussed several years ago
here on NP. 



nikol


Total Posts: 915 
Joined: Jun 2005 


Peters made an update which is storming (weak) brains.
"The ergodicity problem in economics". https://www.nature.com/articles/s4156701907320
@ronin
Very concise summary, however, Peters' example does not cover sentiment. Or scaling position is just consequence of that? 





There are some pretty entertaining Twitter debates between Ole Peters and various economists and quants about "Ergodicity Economics". Peters and his cohorts come off as mathematically competent but crankish... just way too much pure quackery. In that Nature article, this section in particular is bizarre:
"...My second criticism [of Expected Utility Theory] is more severe and I’m unable to resolve it: in maximizing the expectation value — an ensemble average over all possible outcomes of the gamble — expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes (the other members of the ensemble)."



