Non - Randomness Stock Market Price Model

is total variation of the function ( )

. Functions ( )

The proof is trivial if the following representations are used for these functions

Multiplication of the equalities of the lemma defines a hyperbola, hence the name of the property.

Further, on one hand we have

From the above we obtain total variation [ ]

(

Consider the first term. The set of discontinuities of the function ( )

≤ Δ -<ε . This way the set of elementary segments is at most countable. For the variation to be bounded there needs to exist a finite set of elementary segments that would, in a different from the classical understanding sense, cover segment T . Let’s represent the oscillation over the elementary segment in the form of ( )

ρ can be called density of oscillation at the elementary segment. We can state that the set of discontinuities at the elementary segment is at most countable and oscillation is bounded by λ . Over segment T the following condition holds true ( ) ( )

where 0 1 ρ ε < < . Almost all sums k ρ fulfill this condition. For these sums with precision of up to number ρ ε λ the following equality is true ( ) ( )

and the total variation is bounded because

Consider the second term in (1.2). This term complements the total variation of function with the help of the rules of transition between the neighboring oscillations; the rules, in turn, are dictated by the definition for the least upper bounds of the number set. The rules are explained by the diagrams A and B .

k

With that said, formula (1.2) can be written as follows ( )

Here the parameter

is the average density of oscillations over T . The parameter

is the relative deviation of density of oscillation at the ends of T . The parameter ( )

is the average shift of oscillations relative to each other over T . We can say that the parameters characterize non-uniformity (a different density) and anisotropy (prescribed direction for the transitions) of the target space of the function ( )

We combine them into one parame-

Their sum is a natural number, so

and it is also an even number, so To save the number of oscillations for the general case of 0

The range of values for the differences will be determined in this case by the set

This inequality gives a hint for the substitution of variable; we put

where 1 α -< < and . n ∈ 7. The Discussion. There are two special cases. In the case of 0 α = we have the value of equal to

1 2 . In the case of 1 α → the value of ( ) 0 P ζ ≤ may be equal to 0 or to depending on the sign of 1 α . In fact the value of α depends of the frame of reference in which the observer is located. We will call the frame of reference with ζ ′ variable the absolute frame of reference (AFR) and with ζ variable the relative frame of reference (RFR).

One may have noticed that the Model is very similar to probability models. that it is a nonlinear function of ζ . With the help of the Taylor approximation of order two and keeping in mind that for heavy tails ( ) ( )

We find the constants and by using the fact that AFR is invariant to and the approximation we have the equations

G. Soros. The Alchemy of Finance. John Wiley & Sons, Inc. NY, 1987.

Р. Бэр. Теория разрывных функций. Перевод с фр. и ред. А. Я. Хинчина. ГТТИ, М.-Л., 1932.