Rate of price change, how to calculate - page 4

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avtomat:

We can't be so sure in principle, simply because there's only one realization of a process. So the notion of ergodicity has no practical value here.

I don't quite agree. We can evaluate ergodicity as a binary factor (is-no) just like any other process characteristic.

For a stationary process the ergodicity hypothesis is quite natural, for a non-stationary process it is a very strong statement to take for granted. Therefore the first step in checking for ergodicity may be to check for stationarity of a part of the time series (or some transformation of it, why not), or to identify a part where the series can be considered stationary with some certainty. Note that it is possible to do this by one realisation at a time. Further, if we were able to divide the series into ergodic sections, we can apply statistical methods on each of them without overstepping the boundaries, at least with some certainty. That seems to me to be better than nothing.

 
alsu:

I don't quite agree. Ergodicity as a certain binary factor (is-no) we can evaluate just like any other process characteristic.

For a stationary process the ergodicity hypothesis is quite natural, for a non-stationary process it is a very strong statement to be taken on faith. Therefore the first step in testing for ergodicity may be to check for stationarity of some part of the time series (or some transformation of it, why not), or to identify a part where the series can be considered stationary with some certainty. Note that it is possible to do this by one realisation at a time. Further, if we were able to divide the series into ergodic sections, we can apply statistical methods on each of them without overstepping the boundaries, at least with some certainty. That seems better than nothing to me.


I didn't need that hypothesis (c).
.
.
But since you find the property of ergodicity to be necessary_important_useful, the relevant question is: How do you exploit this "ergodicity"?
 
avtomat:

But since you find the property of ergodicity to be necessary/important/useful, the relevant question is: How do you exploit this "ergodicity"?

As said above, the exploitation of the hypothesis consists in 'trusting' various kinds of time averages on ergodic plots and 'distrusting' on non-ergodic ones... in a kind of generalised sense, so to speak.

More specifically, we can give the following example of incredulity: if I

(a) Received a signal for input using some kind of time averages and the hypothesis that they can replace the deterministic component, i.e. the ensemble average,

b) and at the same time I have information that the process was essentially non-stationary/non-ergodic in the analysis section,

then I do not trust such a signal.

 
alsu:

It is not all that straightforward. The article from the handbook applies only to differentiable processes, while stochastic processes, i.e. those with a random component, do not formally belong to such processes: the limit dS/dt does not exist, hence there is no derivative. As stated above, the price can "wiggle" at any small interval of time, and we cannot get inside this interval for purely technical reasons.

That's why I think the question has a non-trivial meaning.

Why is there no limit? A tick is a limit. So we divide the value of a tick (change per tick) at the moment of its occurrence by the time since the previous tick. Dimension is point/second. There is no more limit))

Whether to average or not depends on the specific task and can be deduced by testing

.

 
TSB

Ergodic hypothesis

The ergodic hypothesis (from Greek érgon - work and hodós - path) in statistical physics consists in the assumption that the time-average values of physical quantities characterising a system are equal to their statistical mean values; serves to substantiate statistical physics. Physical systems for which Eg is valid are called ergodic. More precisely, in classical statistical mechanics of equilibrium systems E. g. is the assumption that the time averages of functions depending on coordinates and momenta of all particles of the system (phase variables), taken along the trajectory of the system as points in the phase space, are equal to the statistical averages on the uniform distribution of phase points in a thin (in the limit infinitely thin) energy layer near the constant energy surface. Such a distribution is called a microcanonical Gibbs distribution.

In quantum statistical mechanics, E. g. is the assumption that all states in the thin energy layer are equally probable. E.g., therefore, is equivalent to the assumption that a closed system may be described by the microcanonical Gibbs distribution. This is a basic postulate of equilibrium statistical mechanics because the canonical and large canonical Gibbs distributions (see Gibbs distribution and microcanonical ensemble) can be derived from the microcanonical distribution.

In a narrower sense, E. g. is the assumption put forward by L. Boltzmann in the 1970s that the phase trajectory of a closed system passes through any point of the constant energy surface in the phase space with the course of time. In this form the Eg is wrong because the Hamilton equations (see the canonical equations of mechanics) uniquely define a tangent to the phase trajectory and do not allow its self-intersection. Therefore, instead of the Boltzmann EH, the quasi-ergodic hypothesis was proposed in which it is assumed that the phase trajectories of the closed system approach any point of the constant energy surface as close as possible.

Mathematical ergodic theory studies under what conditions the time averages for dynamical systems are equal to statistical averages. Such ergodic theorems have been proved by American scientists J. Birkhof and J. Neumann. Neumann's ergodic theorem states that a system is ergodic when its energy surface cannot be divided into such finite regions that if the initial phase point is located in one of them, its entire trajectory will remain entirely in that region (the so-called property of metric intransitivity). Proof that real systems are ergodic is a very complicated and unsolved problem.

Lit.: Uhlenbeck J., Ford J., Lectures in Statistical Mechanics, translated from English, M., 1965, pp. 126-30; A. Y. Hinchin. Ya., "Mathematical Foundations of Statistical Mechanics", M.-L., 1943; Ter-Har D., Foundations of Statistical Mechanics, translated from English, Wiley Physical Science, 1956, vol. 59, в. 4, т. 60, в. 1; Arnold V. J., Avez A., Ergodic problems of classical mechanics, N. Y., 1968.

D. N. Zubarev.

.

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Very important and very strict (!!!) conditions of applicability of the ergodicity hypothesis are (1) closure of the system and (2) equilibrium of the system.

Neither of these conditions is satisfied by the market.

1) It is an open system.

2) It is a highly non-equilibrium system.

Methods for studying open non-equilibrium systems do not use the ergodicity hypothesis. (And they do not need such a hypothesis.)

 
avtomat:

Very important and very rigid (!!!) conditions of applicability of the ergodicity hypothesis are (1) closedness of the system

No. The paper describes the ergodicity condition for a closed system, not closedness as a condition. Therefore

1) The market is an open system.

is not an obstacle to ergodicity. The other is,

(2) Equilibrium of the system.

This condition is essential, but the assertion

2) Market is a highly non-equilibrium system.

is not always true. There are areas of equilibrium, or areas that can be reduced to equilibrium by a simple transformation (e.g. subtracting demolition, accounting for seasonality, etc.). This is exactly what I was talking about.

Otherwise, of

Methods for studying open non-equilibrium systems do not use the ergodicity hypothesis. (and do not need such a hypothesis)

follows the impossibility to apply the apparatus of matstatistics to the market in principle, as it substantially relies on the ergodicity hypothesis.


By the way, statistical physics needed the ergodicity hypothesis in order to justify the application of mathematical statistics, without this hypothesis all statistical calculations at least for gas, at least for the market are tantamount to shamanism.

 

Just in case, a counter-example.

A stationary random process is fed to the input of a linear differential filter. The output is also a stationary process.

We have:

1) the system is open

2) ergodicity hypothesis is satisfied, as all time averages are obviously equal to the population mean - expectation, variance, etc., if only they exist.

 
Then the concept of "piecewise" ergodicity should be introduced for the market. In fact, various "continuers" of the graph based on the search for similar plots in the past are trying to carry out this principle unconsciously (or maybe consciously). Although, in fact, when selecting by literal "similarity" the statistics is too weak for reasonable continuation. Some more abstract criteria are needed. Division into flops and trends is probably able to provide statistics, but the problem is with the division criterion :).
 
alsu:

Just in case, here's a counterexample.

A stationary random process is fed to the input of a linear filter - a differentiating link. The output is also a stationary process.

We have:

1) the system is open

2) ergodicity hypothesis is satisfied, as all time averages are obviously equal to the population mean - expectation, variance, etc., if only they exist.


This is a bad counterexample. It's very limited.

As an example, consider a more appropriate model for our case: Some finite volume of a compressible viscous fluid, with a bounded surface, and in motion -- a process accompanied by mechanical work, heat exchange with the external environment, conversion of mechanical energy into heat.

The calculations are more complicated, but much more interesting.

 
avtomat:


This is a bad counterexample. Very limited.

As an example, consider a more appropriate model for our case: Some finite volume of a compressible viscous fluid, with a bounded surface, and in motion -- a process accompanied by mechanical work, heat exchange with the external environment, conversion of mechanical energy into heat.

The calculations are more complicated, but much more interesting.


The question is: "Can you even describe the quadratic trinomial?

The answer is, 'No, I can't even imagine it'.

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