In this research the technology of complex Markov chains is applied to predict financial time series. The main distinction of complex or high-order Markov Chains and simple first-order ones is the existing of aftereffect or memory. The technology proposes prediction with the hierarchy of time discretization intervals and splicing procedure for the prediction results at the different frequency levels to the single prediction output time series. The hierarchy of time discretizations gives a possibility to use fractal properties of the given time series to make prediction on the different frequencies of the series. The prediction results for world's stock market indices is presented.
Successful modeling and prediction of processes peculiar to complex systems, such as ecological, social, and economical (ESE) ones, remain one of the most relevant problems as applied to the whole complex of natural, human and social sciences ( [1][2][3][4][5][6]).
The diversity of possible approaches to modeling such systems and, usually, more than modest success in the dynamics prediction, compel us to look for the reasons of failure, finding them not only in details, but also in the axiomatics, which relates to problem statement, chosen modeling methods, results interpretation, connections with other scientific directions.
With the appearance of quantum mechanics and relativity theory in early twentieth century new philosophical ideas on physical values, measuring procedures and system state have been established, the ones that are completely different from Newtonian notions [7,8].
For more than 70 years basic concepts of classical and neoclassical economic theories have been discussed by leading scientists, generating new approaches [9]. The general systems theory has acquired recognition in the middle of the 20th century giving way to development of the new, systemic, emergent, and quantum in essence approach to investigation of complex objects, which postulates the limited nature of any kind of modeling and is based upon fixed and closed system of axioms [10].
However, the development of this new philosophical basis of ESE systems modeling is still accompanied with numerous difficulties, and new principles are often merely declared.
Current research is devoted to investigation and application of the new modeling and prediction technology, suggested in [11,12], based on concepts of determined chaos, complex Markov chains and hierarchic (in terms of time scale) organization of calculating procedures.
SUBJECT Prediction of financial-economic time series is an extremely urgent task. Modern approaches to the problem can be characterized by the following directions: 1) approximation of a time series using an analytical function and extrapolation of the derived function towards future -so-called trend models [13]; 2) investigation of the possible influence various factors might have on the index, which is being predicted, as well as development of econometric or more complicated models using the Group Method of Data Handling (GMDH) [3,14]; 3) modeling future prices as the decisions-making results using neuronal networks, genetic algorithms, fuzzy sets [14][15][16].
Unfortunately, these techniques don’t produce stable forecasts, what can be explained by complexity of the investigated systems, constant changes in their structure. Although we are trying to join these directions in one algorithm, it is the latter option that we prefer, with it consisting in creating a model adequate to the process generating a price time series [17]. This very approach gives a chance to approach the complexity of the system, which generates the observed series, develop the model and use its properties as the prognosis.
Assume the time series is set by a sequence of discrete levels with constant step of time sampling ∆t. We need to generate variants of the time series continuation (prognosis scenarios) according to the relations between the sequences of absolute and relative changes discovered with the help of complex Markov chains.
Another peculiar feature of ESE systems, apart from complexity, is a memory, including the long-term one, as well as nonlinear and unstable nature of interactions and components, which makes it harder to predict their future behavior.
Unfortunately, mathematical models based on differential equations have no memory (there is no aftereffect), while for models with memory, where integral interrelations are used, it is not always possible to take into account nonlinearity (the integration procedure is linear by definition).
In reality, in the Cauchy problem future systems behavior is defined by its initial state and doesn’t depend on the way the system reached its current state. However, it is hardly true that future behavior of a real socio-economic or socio-ecological system can be predicted by giving an immediate time “slice” of a variables set that describe its state.
Let us consider possible ways to take into account past events while modeling ESE systems’ dynamics, which goes beyond the boundaries of classical differential and integral equations.
Functional differential lagging equation can serve as a simple example of the dynamic model with memory, where present time is defined by the state variable x(t) and depends on the past state x(t -τ ) with constant time lag τ = const:
where f (x) is the known function, with initial conditions being set for the half-interval t 0 -τ ≤ t < t 0 by the function φ(t):
Given the 2 equation 1 has the only solution, defined by recurrent ratios:
Using Dirac delta function, as defined by ratios:
we can formally rewrite equation 1 in the integral form
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