The St Petersburg phenomenon. The paradoxes of probability theory. - page 18

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Aleksey Nikolayev:

The position created by any system is a piecewise constant function of time. On each such piece, the capital increment equals the product of the constant (volume) by the price increment. Therefore the expectation of capital gain is equal to the product of this constant by the expectation of price gain, which is zero for SB without a trend.

Of course, in the general case it is much more complicated as we are talking about conditional expectation of increment, but for SB (by definition) it is the same as the conventional one.

Oleg avtomat:

2) Please give us a link to this rigorous mathematical fact, so that together we can take a look and see the full picture, and not just the dry residue.

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Question: How far will a particle move away from the initial position when a given time has elapsed? Einstein and Smoluchowski solved this problem. Let us imagine that we divide the allotted time into small intervals, say, one hundredth of a second, so that after the first hundredth of a second the particle moved to one place, in the second hundredth of a second it moved further, at the end of the next hundredth of a second it moved further, etc.Naturally, after the expiration of one hundredth of a second the particle does not "remember" what happened to it before. In other words, all collisions are random, so each successive "step" of the particle is completely independent of the previous one. This is reminiscent of the famous problem about the drunken sailor who leaves the bar and takes a few steps, but is bad on his feet, and takes each step somewhere sideways, randomly. So where does our sailor end up after a while? All that can be said is that he is probably somewhere, but this is completely uncertain. What will be the average distance from the bar where the sailor will end up? Theaverage square of the distance from the origin is proportional to the number of steps.Since the number of steps is proportional to the time allocated to us by the conditions of the problem, the average square of the distance is proportional to the time.

However, this does not mean that the average distance is proportional to time. Paradox. If the average distance were proportional to time, then the particle would be moving at a perfectly constant speed. The sailor is undoubtedly moving forward, but his movement is such that the square of the mean distance is proportional to time. This is the characteristic feature of random walks.

http://sernam.ru/lect_f_phis4.php?id=15

This begs the question, what is the MO equal to?

 
Олег avtomat:

You may not have noticed, but that is exactly what I am offering you - aself-calculation check:

But you are not willing to doa self-calculation check.

In case of SB it will be just a check of quality of pseudorandom number generator used, and in a very non-optimal way. Although, sometimes check TC for SB is not meaningless - for example, when evaluating the result of its optimization.

 
Aleksey Nikolayev:

In the case of SB it will just be a test of the quality of the pseudorandom number generator used, and in a very non-optimal way. Although, sometimes it is not unreasonable to check TC for SB, e.g. when evaluating the result of its optimisation.

A lot depends on the MF generator, but not everything.

 
Novaja:

Oleg avtomat:

2) Please give us a link to this rigorous mathematical fact, so that together we can look and see the full picture, and not just the dry residue.

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Question: How far will a particle move away from its original position when a given time has elapsed? Einstein and Smoluchowski solved this problem. Let us imagine that we divide the allotted time into small intervals, say, one hundredth of a second, so that after the first hundredth of a second the particle moved to one place, at the end of the second hundredth of a second it moved further, at the end of the next hundredth of a second it moved further, etc.Naturally, after the first hundredth of a second the particle does not "remember" what happened to it before. In other words, all collisions are random, so each successive "step" of the particle is completely independent of the previous one. This is reminiscent of the famous problem about the drunken sailor who leaves the bar and takes a few steps, but is bad on his feet, and takes each step somewhere sideways, randomly. So where does our sailor end up after a while? All that can be said is that he is surely somewhere, but this is completely uncertain. What will be the average distance from the bar where the sailor ends up? Theaverage square of the distance from the origin is proportional to the number of steps.Since the number of steps is proportional to the time allocated to us by the conditions of the problem, the average square of the distance is proportional to the time.

However, this does not mean that the average distance is proportional to time. Paradox. If the average distance were proportional to time, then the particle would be moving at a perfectly constant speed. The sailor is undoubtedly moving forward, but his movement is such that the square of the mean distance is proportional to time. This is the characteristic feature of random walks.

http://sernam.ru/lect_f_phis4.php?id=15

This begs the question, what is the MO equal to?

The expectation of the mean square of the offset is positive (because the random variable is positive). Expectation of the bias is zero (in case of symmetric walking).

[Deleted]  
Aleksey Nikolayev:

In the case of SB it will just be a test of the quality of the pseudorandom number generator used, and in a very non-optimal way. Although, sometimes checking TC for SB is not meaningless - for example, when evaluating the result of its optimization.

The wall of incomprehension...

Conduct an experiment, it is not difficult. And the existing wall of incomprehension, if not collapsed immediately and finally, will be shaken very substantially.

 
Олег avtomat:

A wall of incomprehension...

I call it another way - understanding the basics of probability theory.

[Deleted]  
Aleksey Nikolayev:

I call it another way - understanding the basics of probability theory.

https://www.mql5.com/ru/forum/70676#comment_2153093

К проблеме неопределённости.
К проблеме неопределённости.
  • 2016.01.03
  • www.mql5.com
Рынок как целое -- система детерминированная.
 
Novaja:

Your level of knowledge is certainly high, add a little more observation and you have an ideal))

Do you also think it is possible to make money from SB?
 
TheXpert:
Do you think you can make money on SB as well?

Why can't you? This paradox:https://www.mql5.com/ru/forum/285122/page7#comment_9131383 proves that the probability of winning when the original decision is reversed is on your side.

 
Олег avtomat:

Do the experiment, it is not difficult, and the existing wall of incomprehension will be shaken, if not immediately and completely.

A simple model for a buy and hold system on SB in R:

c<-rep(0,1000)

for (i in 1:1000) c[i]<-sum(rnorm(10000))

m<-mean(c); s<-sd(c)

m/s #  коэффициент Шарпа

results with multiple runs:

0.01911776

-0.003165045

0.04062785

-0.003669073

Not sure if you can see anything other than what probability theory predicts here (regardless of the level of knowledge and observation)

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