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Jurassic


Total Posts: 235
Joined: Mar 2018
 
Posted: 2019-03-06 12:15
What does it mean for financial markets to be ergodic or non-ergodic processes? Can anyone explain this without having to go into the mathematics too deeply.

Maggette


Total Posts: 1129
Joined: Jun 2007
 
Posted: 2019-03-06 13:43
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 birds-eye 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"

Ich kam hierher und sah dich und deine Leute lächeln, und sagte mir: Maggette, scheiss auf den small talk, lass lieber deine Fäuste sprechen...

ahd


Total Posts: 4
Joined: May 2017
 
Posted: 2019-03-06 14:10
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: 432
Joined: Dec 2008
 
Posted: 2019-03-06 14:25

Can the markets be both reflexive (Soros-sense) and ergodic?

Maggette


Total Posts: 1129
Joined: Jun 2007
 
Posted: 2019-03-06 14:45
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...

gmetric_Flow


Total Posts: 14
Joined: Oct 2016
 
Posted: 2019-03-06 14:57
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: 457
Joined: May 2006
 
Posted: 2019-03-06 16:15
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: 235
Joined: Mar 2018
 
Posted: 2019-03-06 22:27
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
 
Posted: 2019-03-07 21:10
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 sigma-algebra 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 sigma-algebra comes here: Empty set Ø and the universe Omega share that they are both invariant under pre-images: Pre-image T^(-1)(Ø)=Ø and pre-image of T^(-1)(Omega)=Omega. The invariant sigma-algebra 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 pre-imeage 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 non-ergodic 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...

athletico


Total Posts: 957
Joined: Jun 2004
 
Posted: 2019-03-16 14:46
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 time-averages 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 time-averages. The ergodicity argument seems to be more general and widely-applicable in his view.

One of his papers was discussed several years ago

here on NP.
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