=Paper=
{{Paper
|id=Vol-2845/Paper_29.pdf
|storemode=property
|title=Mathematical Models of Pseudorandom Processes Behavior for Nonlinear Dynamical Systems
|pdfUrl=https://ceur-ws.org/Vol-2845/Paper_29.pdf
|volume=Vol-2845
|authors=George Vostrov,Andrii Khrinenko
|dblpUrl=https://dblp.org/rec/conf/iti2/VostrovK20
}}
==Mathematical Models of Pseudorandom Processes Behavior for Nonlinear Dynamical Systems==
https://ceur-ws.org/Vol-2845/Paper_29.pdf
Mathematical Models of Pseudorandom Processes Behavior for
Nonlinear Dynamical Systems
George Vostrov, Andrii Khrinenko
Odessa National Polytechnic University, Shevchenko av., 1, Odessa, 65044, Ukraine
Abstract
This paper considers the processes in maps, which are examples of nonlinear dynamical
systems. Analysing dynamical systems, it is necessary to take into account and analyze
properties of iterative functions that determine the length of nonrepetitive iterative process. It
is shown that not only properties of functions, but also properties of numbers from the
considered functions domain influence the nonlinear maps behavior. Also this work consider
nonlinear maps as a background in the analysis of financial data and complex dynamical
systems, such as stock exchange and economics.
Keywords
Chaos, randomness, nonlinear maps, financial analysis.
1
1. Introduction
Modeling of random and pseudorandom processes in nonlinear dynamical systems based on
randomized algorithms of cyclic trajectories of fixed points requires formation of information
technologies that would include a wide range of mathematical methods for predicting occurrences of
such trajectories and pattern recognition for their behavior that depends on previous states. It is
necessary to take into account the random nature of the depth of transition from one state or sequence
of states to another state. An effective solution to this problem is provided that there are semantic
databases that allow to take into account the experience of preliminary analysis of nonlinear dynamic
systems behavior under different conditions. Estimation methods are required to define true
randomness or pseudorandomness if so-called chaotic behavior observed during number sequence
generations. Algorithms for analyzing such behavior increase the range of approaches that can be
used to effectively solve various problems in areas such as modeling of dynamical systems, functional
analysis, function theory, cryptography and others. The obtained sequences are widely used in the
Monte-Carlo method where sequences of pseudorandom points are used to calculate multidimensional
integrals [1], in the theory of machine learning to obtain training and test sequences [2] and others.
Various methods of number sequence generation are based on chaotic processes and the next question
arises: “What processes can be considered as chaotic?” There is no precise and constructive axiomatic
definition of the concept of randomness at this moment. In Kolmogorov axioms, random sequences
were left outside the theory and only general approaches to the definition of randomness were
proposed, such as von Mises's approach. In this case, even if a method of sequence generation is used,
there is a problem of uncertainty of the degree of approximation to the real randomness, because a
truly random sequence is infinite and cannot be generated for real-world problems. Numerical
sequence generators provide pseudorandomness with a certain degree of approximation to a given
distribution law. Thus, the construction of random number generators is a way to define the formal
concept of randomness, which is important and necessary in modern probability theory, stochastic
process theory and others. Since a truly random sequence is a mathematical model that is a
IT&I-2020 Information Technology and Interactions, December 02–03, 2020, KNU Taras Shevchenko, Kyiv, Ukraine
EMAIL: vostrov@gmail.com (G. Vostrov); khrinenko@stud.opu.ua (A. Khrinenko)
ORCID: 0000-0003-3856-5392 (G. Vostrov); 0000-0001-6000-2102 (A. Khrinenko)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
308
completely unpredictable and therefore non-periodic infinite sequence, the problem of generating a
pseudorandom sequence (PRS) is formulated [3]. Separate branch of science, which is called
deterministic dynamical systems also refers to the problem of chaos. The basis of this branch is the
analysis of the dynamics of iterative fixed points using recursive functions. In this case, iterative
cycles or orbits are considered as deterministic chaos [7], because they depend on the initial
conditions, and also show irregularity [4]. Deterministic dynamical systems arise as a result of
approximation of complex processes from the physical and mental world and can be used as
generators of pseudorandom sequences. Another advantage of dynamical systems is that the required
level of approximation to randomness can be achieved by a combination of several algebraic
equations. In the case of PRS generators and dynamical systems, it is necessary to take into account
and analyze properties of iterative functions that determine the length of nonrepetitive iterative
process, which is one of the main properties of generators. In this case, the properties of the set of
numbers on which these iterative functions are defined draw much less attention. There is a direct
connection with number theory and above-stated problem. Prime numbers are of considerable interest
because factorization can not be applied to them while usage of large compound numbers does not
provide required cycle length. In many practical problems, the initial value is determined by random
selection of a prime number of large dimension, but some prime numbers also do not allow obtaining
of desired cycle length and approaching truly random sequence. The numbers of this class include of
Fermat, Mersenne, Wagstaff numbers and their generalizations [1]. In this case, neighboring prime
numbers provide a cycle length corresponding to the dimensionality of the number and meet the
requirements to the resulting sequence. Thus, insufficient analysis of the selected numbers can lead to
significant errors in the obtained sequences and incorrect conclusions.
2. Analysis of iteration processes in nonlinear maps
To analyze above-formulated problem, this paper considers the processes in maps, which are
examples of nonlinear dynamical systems. By process it is meant a function F, that present the final
sequences (words) in words such that if for the word x F (x) is defined and y x, then F ( y ) also
defined and F(y) F(x) . Let - some sequence. The process F will be reapplied as long as
possible. As a result, it will be obtained fragments of some new sequence P - the result of application
F to , ie P F (). Nonlinear maps of the following classes are considered: “Tent”, “Asymmetric
tent”, “Sawtooth” and multiplicative order map. It is important to note that maps (1,2,3) are
continuous functions, while maps (4) are defined only on the set of integers. The choice of these maps
allows us to consider and analyze the chaotic processes that are also observed in complex dynamical
systems.
2 x n , x n 1 2 x n , x n 1
4 2
t 1 ( x n ) x n 1 (1) t 2 ( x n ) x n 1 (2)
1 2 xn , xn 1 4
1 xn , xn 1
2
2 x n , x n 1
2
t 3 ( xn ) xn1 (3)
2 xn 1, xn 1
2
It should be noted that computer systems in calculations operate with numbers in binary form and
of limited length, while mathematics operates with numbers of infinite length and these circumstances
lead to errors during operations with fractional numbers, which ultimately creates a problem of
reliability and gives false conclusions about the processes in dynamic systems due to the fact that one
of the properties of any dynamic system is the sensitivity to initial conditions [4]. In order to minimize
this circumstance, the transition to integer maps is performed, which are presented as follows:
2 xn , 4 xn p 2 xn , 2 xn p
xn1 (1) xn1 (2)
p 2 xn , 4 xn p p xn , 2 xn p
309
2 xn , 2 xn p
xn1 (3) xn1 4 xn (mod p), (4)
2 xn p, 2 xn p
where p – prime number. The graphs representing maps (1,2,3) in the interval [0,1] are shown in
Figures 1, 2, 3, respectively, and the graph 4 gives the definition of the map (4) on the set of integers.
In the presented figures, the graphs of the maps intersect the diagonal y x at some points, which are
fixed periodic points. It should also be noted that the map (1) is algebraically congruent to the map (4)
on the set of integers, i.e. the cycle lengths for all prime numbers coincide. And map (2) does not
satisfy in the general case to Fermat's little theorem.
Despite the simplicity of the above maps, their iterative cycles have properties that confirm the
above statements. According to them, the structure of iterative cycles is determined not only by the
properties of the maps themselves, but also by the properties of the numbers on which these maps are
based and which have a significant impact on the structure and can significantly change it. The
presented nonlinear maps allow to divide the set of primes p into a system of classes based on the
length of the iterative process for given prime numbers [5]. Since such an internal structure is an
exception, the question of introducing a certain degree of similarity of internal structures arises. For
example, for the prime number 649657, Figures 5 and 6 show the internal structure of iterative
processes for maps, where the dotted line shows the resulting sequences and the solid line shows the
internal parts within the sequences that give the maximum value of the correlation coefficient.
It is worth noting that the choice of a prime number of large dimension does not allow to avoid
sequences with a simple internal structure and internal exponential and periodic components, which
are indicated by the green region on the Figure 7. At the same time, the previous and next prime
numbers generate sequences for which the length of the period corresponds to the dimension of the
number and may indicate a better approximation to the conditions for pseudorandom sequences, but
also have periodic components observed in places of compaction on the Figure 7.
Figure 1. 1st and 2nd iteration for the map 1
Figure 2. 1st and 2nd iteration for the map 2
310
Figure 3. – 1st and 2nd iteration for the map 3
Figure 4. – 1st and 2nd iteration for the map 4
Considering prime numbers, special attention is drawn to prime numbers of a special form, such as
generalized Gaussian Mersenne numbers - prime numbers p* 2 p 1 for which the number
((1 i) p 1)((1 i) p 1) and p are also prime numbers. It is possible to identify sequences that have a
simple structure with exponential components that are repeated with a certain frequency and slightly
different in amplitude. These numbers completely violate the conditions of randomness, which are
essential for PRS. It is proved that such an internal regular structure is characteristic to Mersenne,
Wagstaff numbers and their various generalizations. Thus, Figure 8 shows the structure of PRS for the
generalized Gaussian Mersenne number.
3. Synergetic behavior of nonlinear dynamical systems of stock exchanges
a. Wave fluctuations in nonlinear processes of stock exchanges
The processes of investing financial resources in the world economy for the purposes of specific
countries or their regions are multidimensional random processes (t ) ( 1 (t ),, n (t )) that are a
dynamical system. Each component i (t ) in (t ) is a multidimensional random process
i (t ) g i ( i1 (t ), , im (t ),) of different dimensions.
In all cases of examination of a certain random process of formation of security quotations it is not
possible to build a mathematical stochastic model of a random vector (t ) and its individual
components. In real life, stock market processes are registered using modern computer information
technologies, which allow to calculate various parameters, such as prices of stocks, options and other
securities at the opening and closing of the stock exchange and the number of their sales.
311
Figure 5. Sequence structure generated by prime number 649657
312
Figure 6. Sequence structure generated by prime number 26295457
313
Figure 7. Nonlinear dynamics by prime 160465519 for map 1
Figure 8. PRS based on Gaussian Mersenne prime
The dynamics of price fluctuations is represented by time series in which time can be recorded
with high accuracy (up to a millionth of a second). Time series are registered for all bidders in the
markets. The time series X (t1 ),, X (t n ), for each company is essentially a statistical model of a
random process (t ) .
The main mathematical tools for studying self-organized nonlinear dynamic processes of stock
markets are the analysis, processing and forecasting of dynamic processes for all market participants
on the basis of time series. Next, it needs to explore some mathematical features of each one-
dimensional time series. It is proved that the dynamics of price fluctuations is cyclical [1]. The study
of the structure of cycles of price fluctuations is an important and very complex problem of statistical
analysis of time series [3].
It is established that the one-dimensional time series of share quotations and options on the stock
markets X(t) is a mixture of at least three random processes. The term "mixture" emphasizes that the
components of the mixture are not independent random variables at all, but the relationship between
them is nonlinear, and they can be uncorrelated in individual time intervals, where nonlinear
dependence has a significant meaning of behavior with a particular strategy. The three underlined
314
components X(t) are related to three random processes with different evolutional mechanisms.
Computer analysis of time series of stock prices allows to establish that the dynamics of time series is
largely determined by the following factors:
1) Economic fluctuations of chaotic deterministic nature are due to regular fixed points, which
can asymptotically approach the general limit [6];
2) Fluctuations of X(t) are in the nature of random processes with independent increments due
to the large number of market participants' interests, which are constantly changing when
buying and selling securities [2];
3) The structure of wave fluctuations is due to the ongoing Elliott wave generation process in
which fluctuation trend always consists of five growth fluctuations associated with price
increases and three decreasing fluctuations due to opposite corrective actions.
In the first case, at any time t i , the time series X (t i ) have a form that corresponds to the logistic
equation X (t n1 ) aX (t n )(1 X (t n )) , which has the following graphical representation.
Figure 9. Graphical representation of Logistic map
This process describes the acquisition of stocks in the interval (t n , t n 1 ) , which leads to the
exhaustion to some extent of this resource at the time t n k (this may be the end of the trading day).
As proposed in [6] for the chaotic nature of logistic map it is proved that it connects with the map
"Tent" which allows to consider in detail the dynamic nature of processes and to investigate the
iterative process of fixed point formation [7,8].
In the second case, based on the fact that the continuous flow of orders on the stock exchange
contains unrelated orders, the model "Geometric Brownian motion" can be used to simplify the
description of the buying and selling process. This approach to stock markets was first proposed by
M. F. M.Osborne [5]. This model, in contrast to the Brownian motion, is everywhere positive and has
a log-normal law of price distribution. Figure 10 shows a representation of the Brownian motion
trajectory for a particle on the 2D plane and a simulation of the geometric Brownian motion process.
Processes of this kind lead to constant fluctuations during one trading day.
However, it follows by randomness that it is impossible to predict the rise or fall of any particular
stock, but due to the law of large numbers it is possible to obtain information about the behavior of
markets in the long run.
In the third case, it is possible to observe the basic law of nature [4], according to which the
development of the price of any stock (option) consists of five fluctuations with increasing and
decreasing its values, but close to a certain value and with the next three stages of price adjustment of
the financial instrument. A graphical representation of the process is shown in Figure 11:
315
Figure 10. Brownian motion simulation
Figure 11. Elliot wave structure
The number of development waves can be more than five, but the number of waves of correction is
always smaller. The construction of the mathematical theory of Elliott waves is associated with the
development of mathematical methods for modeling human decision-making processes in a socio-
dynamic environment. An analysis of the decision-making processes of people (brokers) in the stock
markets confirms that the desire to sell a financial asset at a higher price always requires greater
fluctuations in decisions than decisions related to adjusting prices at some acceptable level. The
structure of Elliott waves agrees with the wave lengths of individual waves at the stage of
development and at the stage of correction. It is established that the number of waves at the
development stage always exceeds the number of waves at the correction stage. This is due to the fact
that the adjustment phase is always associated with minimization of financial losses and maximization
of the possible rate of return.
Analysis of the observed Elliott waves confirms the view that these waves are the result of the
collective intellectual activity of a large number of people with agreed strategies for success, and also
confirms the Keynesian business cycle model [2]. It has been proven that the use of Elliott waves, the
theory of fixed iteration points and random walk processes create a basis for stabilization and
improvement of the efficiency of stock markets as a regulator of financial assets investing in the
developing world economy.
b. Nonlinear dynamical processes in synergetic economics
Analysis of developed and emerging economies of the world suggests that they all function as
synergetic systems. There is still no exact mathematical definition of the synergetic nature of such
processes. The model of synergetic dynamics of interaction of various components of nonlinear maps
describing processes of development and realization of innovative projects is considered.
316
The processes of innovative projects consist of two components. The first stage is the creation of a
new technology for studying the management of economic systems and the introduction of an
innovative development process based on investment in the business plan. This process is
systematically repeated in high-tech companies:
1) new development process;
2) investment in a business plan;
3) implementation of development in the markets
And again, this sequence of stages of innovation is repeated at a new level.
The stability of these processes must be accurately explained from a mathematical point of view.
Research papers [1, 9, 10] contain the results of research aimed at explaining the synergistic nature of
economic development associated with the development of high technology (Microsoft, Google, IBM,
Tesla, etc.). The considerations presented in these works do not always give a full justification of their
synergetic nature.
The authors of [2, 11, 12] first drew attention to the Levy factor, when investment in a project is
sharply reduced, and very quickly allocated to the implementation of a new investment (promising)
project in the same field and the same companies.
This is especially evident in projects related to electric vehicles, solar power plants, system
software, smartphones, information technology and more.
Analysis of nonlinear dynamic processes in such areas allows us to formulate and prove that such
synergetic processes in these companies and the economy as a whole always contain several nonlinear
components of such processes, close to the logistic model of resource consumption, with moments of
restructuring in the form of Levy flight [12 ], ie periodic changes in investment strategy. Levy's flight
is one of the forms of random wandering, the graphic image of which is shown in Figure 12.
Figure 12. Representation of Levy Flight process
Each component performs (realizes) its contribution to the formation of stability. Functions of type
S 1 (t ) are the process of transferring funding from one project to a completely new one. The process
S 2 (t ) is implemented when at some point the implemented innovation project receives corrective
funding, which leads to a decrease in the level of innovation and its gradual decline. Maps S 3 (t ) and
S 4 (t ) describes the single processes of investing financial resources and their depletion. A graphical
representation of these processes is shown in Figure 13.
Undoubtedly, the number of components of these types can be much larger. Analysis of the S
function using its computer modeling and mathematical analysis with the right choice of weight
functions always provides the necessary level of stability for these types of investment management
processes in all areas of developed economies and much less in developing countries. A more
complete mathematical analysis of this approach and the results of computer modeling will be the
subject of a separate publication.
c. Fourier analysis in finance: price formation and Fourier transform
A wide range of real systems demonstrates the properties of randomness, including finance and
stock markets in particular. The question arises: "Does a system like the stock market develop by
317
accident?". The hypothesis of random wandering as the basis of the market was formalized and
popularized in [3]. In order to test the correctness of this hypothesis, it is necessary to determine
whether the results formed in the market are stochastic or deterministic.
To solve this problem, a statistical approach was used, using a set of NIST tests and other
cryptographic tests for randomness. It has been shown that markets are not random because financial
data do not satisfy the properties of randomness, such as chaos, frequency stability and
unpredictability.
Accordingly, it is possible to build methods for forecasting financial data on historical changes in
stock prices and a wide range of financial derivatives. Among the various mathematical tools that can
be applied to finance, the Fourier transform is of particular interest because it makes it possible to
identify patterns or loops in time series data and also provides a real-time pricing method for a
financial instrument such as the option that was considered in the work of Carr and Madan [19].
Figure 14. Representation of investment processes
The very first option pricing model used a geometric Brownian process to model the basic pricing
process. But due to some known shortcomings of this approach, it was proposed to move to the
options of pricing models based on Levy processes.
Therefore, the Fourier transform technique is an effective approach to estimating the option
according to the Levy model, because the Levy process X t can be fully described by its characteristic
function X (u ) , which is defined as the Fourier transform of the density function X t .
To describe the application of the Fourier transform (FT) in the option pricing model, it is
important to establish a definition of the transformation. If we analyze the piecewise-smooth function
f (x) on (,) , then the FT is defined as:
F f (u ) e iuy f ( y )dy (1)
If we consider a sequence of numerical values { x k } as a result of a discrete Fourier transform (DFT)
for another sequence { y j } and is defined as follows:
318
N 1 2 jk
i
y j xk e n (2)
k 0
Here, the Fourier transform can be considered as a projection of the sequence { x k } on the vector
space, because each product is a projection on a single basis vector. Next it is necessary to describe
the Levy process.
An actual process is called a Levy process if it exhibits certain properties: independent increments,
stationary increments, and almost certainly a continuous right map with a limit on the left. The Levy
process can be represented as a combination of linear drift, Brownian process and jump process.
Turning to the application of PF for option pricing, we can describe an alternative formulation of
option pricing, which uses an analytical expression of the characteristic function of the basic process
of asset price using fast Fourier transform (FFT).
To obtain a quadratic-integral function, it was proposed to consider the Fourier transform of the
declining price of the transaction c(k ) , where c(k ) e ak C (k ) for a 0 . Positive values a improve
the integration of the modified transaction value over the negative axis k.
Consider T (u ) as a Fourier transform c(k ), pT (s ) as a function of the density of the underlying
process of the asset price, where c(k ) s log S T and T (u ) as a characteristic function pT (s ) . This
gives:
e rT T (u (a 1)i)
T (u ) e iuk c(k )dk (3)
a 2 a u 2 i(2a 1)u
The transaction price C (k ) can be restored by the inverse Fourier transform, where
e ak iuk e ak iuk
C (k )
2
e T (u)du
0
e T (u)du
(4)
The above integral can be calculated using FFT. And it is important to note that a should be such
that the denominator has only imaginary roots in u , because the integration is performed on the real
value u .
Next, it is necessary to describe the evaluation process for the integral (4). Let's start with choosing
the number of intervals N and bandwidth u . Therefore, the numerical approximation for C (k ) is
given by the formula:
e ak
N iu k
C (k )
e j T (u j )u , (5)
j 1
where u j ( j 1)u, j 1,...., N . The semi-infinite integral domain [0, ) in the integral in (4) is
approximated by a finite integral domain, where the upper limit for in numerical integration is
represented as Nu .
Also, the Fourier variable u is selected at discrete points instead of a continuous representation.
From one set of FFT calculations it is possible to get the price of the transaction (call) for the option
for a set of strike prices.
This gives market analysts the opportunity to record the price sensitivity of the transaction with
different execution prices.
The transaction price is multiplied by the corresponding exponential attenuation coefficient,
converted into a square-integral function, and the Fourier transform of the modified transaction price
is converted into an analytical function of the characteristic function of the registration price (log
price). However, the short maturity causes significant numerical errors, so the time component of the
option should be considered to obtain smaller pricing errors for all execution price levels.
319
4. Conclusion
Summarizing the above information, it should be noted that Fourier transform methods provide an
approach to pricing for financial instruments based on Levy processes, because the analytical
representation of the characteristic function is more accessible and can be obtained using the
previously described approach with high accuracy and efficiency. It is possible to get option prices for
the full range of exercise prices in one set of FFT calculations. And this approach can be further
developed for other derivatives on the Levy model, where the payout function depends on the
volatility of the underlying process.
5. References
[1] R. Crandall and K. Pomerance. Prime numbers: cryptographic and computational aspects, 2nd
ed., Springer, 2005.
[2] V. Balasubramanian, S. Ho, V. Vovk. Conformal Prediction for Reliable Machine Learning:
Theory, Adaptations and Applications, Elsevier, 2014.
[3] A. Shiryayev. Veroyatnost i kontseptsiya sluchaynosti: k 75-letiyu vykhoda v svet monografii
Kolmogorova “Osnovnyye ponyatiya teorii veroyatnostey” [Probability and the concept of
chance: to the 75th anniversary of the publication of the monograph by A. Kolmogorov “Basic
concepts of probability theory”]. Materialy seminara “Matematika i yeye primeneniye”,
Moscow, MIAN, 2009. (In Russian).
[4] M. Hirsch, S. Smale, R. Devaney. Differential equations, dynamical systems, and an introduction
to chaos, Elsevier, 2013.
[5] G. Vostrov and R. Opiata. Effective computability of dynamic system structure of prime number
formation, ELTECS (2017).
[6] A. Sharkovsky. Attraktory trayektoriy i ikh basseyny [Trajectory attractors and their pools]. –
Kyiv: Naukova Dumka, 2013. (In Russian).
[7] G. Shuster. Determinirovannyy khaos: Vvedeniye [Deterministic chaos: Introduction]. – Mir,
1988. (In Russian).
[8] V. Uspenskiy. Chetyre algoritmicheskikh litsa sluchaynosti [Four algorithmic faces of
randomness]. – Moscow, MTSNMO, 2006.
[9] W. Zhang. Synergetic Economics: Time and Change in Nonlinear Economics, Springer, 1991.
ISBN 3540529047.
[10] A. Panchenkov. Econophysics: An Introduction, Printing house “Povolzh'ye”, 2007. ISBN 978-
5-98449-076-4 .
[11] J. Hull. Options, Futures and Other Derivatives, Prentice Hall, 2005. ISBN 0131499084.
[12] E. Peters. Fractal Market Analysis: Applying Chaos Theory to Investment and Economics,
Wiley, 2009. ISBN 0471585246.
[13] M. Osborne. Motion in the Stock Market, Operations Research (1959).
[14] G. Vostrov and A. Khrinenko. Computer modeling of the chaos formation processes in nonlinear
dynamic maps, ELTECS (2017).
[15] E. Peters. Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market
Volatility, Wiley, 1996. ISBN 978-0-471-13938-6.
[16] H. Haken. Synergetics: An Introduction, Springer, 2005. ISBN 3540123563.
[17] R. Mantegna. Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge
University Press, 2007. ISBN 0521039878.
[18] P. Carr and D.B. Madan.Option valuation using the fast Fourier transform, Journal of
Computational Finance (1999).
320