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.
completely unpredictable and therefore non-periodic infinite sequence, the problem of generating a
pseudorandom sequence (PRS) is formulated [
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.
2xn , xn 14 t1 (xn ) xn1 1 2xn , xn 14 (1) t2 (xn ) xn1 2xn , xn 1 2 (2)
1 xn , xn 1 2 t3 (xn ) xn1 2xn , xn 1 2 (3)
2xn 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 [
2xn , 2xn p
xn1
p xn , 2xn p
(2)
2xn , 2xn p
xn1
2xn p, 2xn p
(3)
xn1 4 xn (mod p), (4)
where p – prime number. The graphs representing maps (1,2,3) in the interval [
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 [
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. 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. a.
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 i (t) gi (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. (t) is a multidimensional random process
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 (tn ), 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
onedimensional time series. It is proved that the dynamics of price fluctuations is cyclical [
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
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 [
In the first case, at any time ti , the time series X (ti ) have a form that corresponds to the logistic equation X (tn1 ) aX (tn )(1 X (tn )) , which has the following graphical representation.
This process describes the acquisition of stocks in the interval (tn , tn1 ) , which leads to the
exhaustion to some extent of this resource at the time tnk (this may be the end of the trading day).
As proposed in [
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 [
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 [
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 sociodynamic 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 [
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.
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 [
The authors of [
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 [
Each component performs (realizes) its contribution to the formation of stability. Functions of type S1 (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 S3 (t) and S4 (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.
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
accident?". The hypothesis of random wandering as the basis of the market was formalized and
popularized in [
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].
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) eiuy f ( y)dy
If we consider a sequence of numerical values {xk } as a result of a discrete Fourier transform (DFT) for another sequence {y j } and is defined as follows: (1)
Here, the Fourier transform can be considered as a projection of the sequence {xk } 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) eak 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 ST and T (u) as a characteristic function pT (s) . This gives: T (u) eiuk c(k)dk
erT T (u (a 1)i) a 2 a u 2 i(2a 1)u The transaction price C(k) can be restored by the inverse Fourier transform, where
C(k) e2ak eiukT (u)du eak 0 eiukT (u)du
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:
C(k ) e ak
Nj1 e iu jk T (u j )u, 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. (3) (4) (5)
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.