The Model of Real Data Constructing Using Fractional Brownian Motion

Let’s assume , ,.., 1 , n k t t k = = , 0 , 0 ) 1 ( 1 1 = − − + = t n T k t tk ) ))( ( ( k k t X x ⋅ Φ = is a model of observed time series . ... 1 n x x Let’s call ξ as a basic process of the model. Levy processes with independent stationary increments have been considered as basic for models of time series (particularly financial) [1-4]. The next step in the development of the models is transition to diffusion processes. For example, diffusion model of stock price ) (t S is obtained from the following considerations:

is an observed trajectory, which is describing the stochastic evolution of some dynamic object. Mathematical model of this trajectory is defined as a random process, x(t) = X(t), X( ) is realization of the processξ .
As a rule, we chose as a model random process with known characteristics. Direct use of this definition requires broad classes of these processes. On the other hand, this class includes Gauss and Markov processes. Let's introduce another definition of continuous mathematical models for the observed trajectory ) ; 0 ( ) ( T C x ∈ ⋅ using nonlinear conversion. Let's call ξ as a basic process of the model. Levy processes with independent stationary increments have been considered as basic for models of time series (particularly financial) [1][2][3][4]. The next step in the development of the models is transition to diffusion processes. For example, diffusion model of stock price ) (t S is obtained from the following considerations:

Mathematical model of observed trajectory x (t) is a pair
w is a standard Wiener process, σ is volatility and interest rate, µ is a constant. Then let's propose the equation: Which can be interpreted as a stochastic equation Ito and its solution could be written as a geometric (economic) Brownian motion: For the model (1) of a stock price have been obtained a number of known results, including the Black-Scholes formula for a rational option pricing [5][6][7][8]. The main drawback of Levy processes (and diffusion) is their Markov. Thus, the Markov property: A priori satisfies only the simplest physical phenomena. The absence of impact on processes in biology, economics, climate, etc. looks unconvincing. In this paper we propose a non-Markovian model of the time series.

Selection of the Base Process and its Properties
One of the most popular Markov models of time series is Gaussian random process, and fractional Brownian motion [9][10][11]. The demand of this process is caused by "convenient" properties, which are described below.
Fractional Brownian motion is defined as a Gaussian random process with characteristics: we get a standard Wiener process.
Smoothness of the trajectories of the process is defined by the parameter H: almost all the trajectories satisfy the Holder condition: This generalizes known Levy's result for the Wiener process.

The increments of fbm
t t t t < < < are form a Gaussian random vector with a correlation between the coordinates: For discrete time: We obtain the correlation coefficient: It means that increments are forming stationary (in the narrow sense) sequence [12].
Let's mention some several properties of fBm: 1. Changing time scale is equivalent for changing of "amplitude" of the process:

Law(B H (at))= Law(a H B H (t)),
This equality denotes the coincidence of one-dimensional distributions of the processes: This property is called self-similarity process and it is useful for analysis of time series.
2. Let's put in the formula (2): j=k+n. Then the correlation coefficient: ) , ( So the memory decreasing for increments has a power character; the increments are independence with   The properties of persistence have data which are describing some of the physical processes, such as solar activity [13,14].
In this paper, is selected fractional Brownian motion as a basic process.

The Statistics of Fractional Brownian motion The estimation of Hurst exponent
Let's observe the data: , where the correlation matrix The limit theorems for sequence n y y ,.., 1 were first proved by Peltier for statistics [15].
From (5) is follows consistency estimates of parameters With known H.
Let's introduce the notation: Is a consistent estimator of the parameter H.

Proof
ε Is the canonical Gaussian vector with the following characteristics: With calculating the dispersion Use the formula for integration by parts for the Gaussian measure [19] and get 0 2 σ Is a consistent parameter estimation of σ .
The equalities (6,7) are form the system, from which follows the relation: This proves the statement. The efficiency of proposed estimation method has been tested by numerical experiment [16].

The limit theorems for some statistics
The limit theorems for statistics from increments of fractional Brownian motion have been proved in works of Nourdin I and others [20][21][22][23].
There is a Mean-square convergence: These results allow us to estimate the adequacy of model with the basic process-fractional Brownian motion.  The statistics Q is calculating for difference values of Hurst exponent with step (0, 0.5-1) and:

Construction and Checking the Adequacy of Model
3) The testing of hypothesis T= (statistics n z z ,.., 1 which obtained by transformation (11) of real data are simulated by increments from fractional Brownian motion). The algorithm with known H is the following. Denote The decision about hypothesis T is accepted by comparing the real values of the statistics with their theoretical limit values. Let's determine deviation from the limit values for statistic n A .
The hypothesis T is accepted, if: Where  (Figures 3 and 4).

The Comparative Analysis of Used Models
Let's compare the time series model (approximation of fractional Brownian motion) with known models and estimate the quality of modeling. Note that the choice of the quality criterion is dependent from the type of model.
The values of exchange rate, Banque de France. Let's compare the effectiveness of approximation method with other models for real 1020 data [21]. These methods have been selected, because after using the special tests for statistical data, we've revealed high autocorrelation value and existence of a trend.
Based on analysis of values of the constructed partial autocorrelation and autocorrelation function of data series, the order AR (1) model may be in the range from 1 to 5. The model AR (1) is given: is a random process. Сharacteristics of the adequacy and quality for short-term forecasts for the training sample had the following values:

=10.82
However, its characteristics aren't better than in the previous case (the model with the trend), except for Durbin-Watson statistic (Table 1) Thus, the best is a structural model of the process from all constructed mathematical models, which takes into account explicitly the trend of process and vibrations (MARD=5.78%) [17][18][19]. This is quite a logical result, because the structural models are describing these processes with a higher degree of adequacy than others. As expected, the introduction of a moving average model didn't improve the quality as compared with a simple model AR (1). The value of the Durbin-Watson statistic for the approximation model is more closer to 2 than  the other models, the value of MARD is practically coincides with its value for the AR (5) "plus" cubic trend.
We consider the primary data processing as a calculation of linear approximation of the trend of initial data and obtaining new sequence  (11), the estimation of Hurst exponent by formula (12), checking the quality of approximation by formula (10). The results of calculations are shown in Table 2.

Conclusion
For all examples the approximation has antipersistent character (H<0.5) and it's adequate, if the conditions are satisfied [14].