The 14th INTERNATIONAL SCIENTIFIC CONFERENCE INFORMATION TECHNOLOGIES AND MANAGEMENT 2016 April 14-15, 2016, ISMA University, Riga, Latvia
Pashko A
Simulation of the generalized Wiener process A Pashko* Taras Shevchenko University of Kyiv, Glushkova Str. 4d, 03187 Kyiv, Ukraine *Corresponding author’s e-mail:
[email protected]
Abstract This article analyzes the modeling methods of the generalized sub-Gaussian Wiener process. Modeling algorithms are based on the representation of generalized Wiener process in a series of random and stochastic integral. Number of components in the model is selected based on the accuracy of the simulation. Keywords: simulation, generalized Wiener process, sub-Gaussian random model, accuracy and reliability of model
1 Introduction ak
Statistical models of Wiener Processes are used in many applications, such as the calculation of integrals for the Wiener process, the numerical solution of stochastic differential equations in problems of actuarial mathematics. In problems of statistical modeling to evaluate the accuracy of simulation, usually used the assessment points, assessment of weak convergence of distributions and assessment of accuracy and reliability in function spaces. As a model considered spectral representation of random processes in a random series and stochastic integrals.
C
X k , Yk
Let
(1)
.
xk xk hkx ,
bk bk hkb ,
The
process
modeling
as
S (t , M ) ak sin xk t X k bk 1 cos yk t Yk . M
k 1
The
modeling
accuracy
(t )
will
be
(t ) W (t ) S t, M . Wiener process with arbitrary index Hurst can be written as integrals
- independent standard Gaussian random
variables, xk - real zero of Bessel functions J ( x) ,
yk - real zero of Bessel functions
ak ak hka ,
yk yk hky
where
X k , Yk
- independent sub centered strictly Gaussian
random variables. Designate the approximate values ak , bk , xk , yk by ak , bk , xk , yk .
Generalized Wiener process can be represented as a series [1]
.
,
random variables with EX k2 EYk2 1 . The correlation function of this process will coincide 1 2 2 2 with R (t , s) t s t s . 2 Let X k , Yk - independent strictly sub-Gaussian
1 2 2 2 t s ts . 2
W (t ) ak sin xk t X k bk 1 cos yk t Yk ,
2 1
yk 1 J ( yk )
EW (t ) 0 in the representation (1) the sequence
W (t ) with zero mean and correlation function
k 1
2 1 sin( )
2C
strictly sub- Gaussian generalized Wiener process with Hurst index 0,1 if W (t ) such as W (0) 0 ,
Let T , , - be some measurable space. Generalized Wiener process with parameter , 0,1 will be called a Gaussian random process
xk 1 J1 ( xk )
, bk
The random process W (t ), t 0, T will be called
2 Algorithms of statistical modeling
R (t , s) EW (t )W ( s)
2C
J1 ( x) ,
44 CM04
Computer Modelling and Information Technologies
The 14th INTERNATIONAL SCIENTIFIC CONFERENCE INFORMATION TECHNOLOGIES AND MANAGEMENT 2016 April 14-15, 2016, ISMA University, Riga, Latvia
A cos(t ) 1 sin(t ) , W (t ) d ( ) d ( ) 2 1 2 1 0 0 2 2
Pashko A
set [0, ] . The process simulation as (2)
M sin(i t ) A M cos(i t ) 1 S M (t , ) 2 1 X i 2 1 Yi , i 1 i 1 i 2 i 2
where ( ), ( ) - independent real standard Wiener E ( ) E ( ) 0
processes,
and
E (d ( )) E (d ( )) d 2
2
where
,
variables,
1
2 1 cos( ) A2 d . 2 1 0 The model is based on an algorithm. Let ВM : 0 0 1... M - a partition of the
X i , Yi
- independent sub-Gaussian random E ( ) E ( ) 0
E (d ( ))2 E (d ( )) 2 d . The modeling accuracy
(t )
and will
be
(t ) W (t ) S M t , .
FIGURE 1 The implementation of the generalized Wiener process on model (1)
FIGURE 2 The implementation of the generalized Wiener process on model (2)
FIGURE 3 The implementation of the generalized Wiener process on model (1)
FIGURE 4 The implementation of the generalized Wiener process on model (2)
Figure 1 and Figure 2 shows the implementation of the generalized Wiener process for 0.2 . Figure 3 and Figure 4 shows the implementation of the generalized Wiener process for 0.5 (Standard Wiener process).
3 Conclusion The behavior patterns received by model (1) and (2) the same. Therefore, any representation can be used for the simulation. Implementation of the model (1) requires the calculation of zeros of Bessel functions with the required accuracy. Model (2) more easy to implement, but it must be a compromise between the range and the number of terms in the model. Estimates for the accuracy and reliability of modeling Wiener process in various function spaces studied in [2-3].
References [1] Dzhaparidze K, Zanten J 2010 A series expansion of fractional Brownian motion CWI. Probability, Networks and Algorithms R0216 [2] Pashko A 2014 Statistical simulation of the generalized Wiener process. Bulletin of Taras Shevchenko National University of Kyiv.
Series Physics & Mathematics 2 180-3 [3] Pashko A 2014 Simulations of standart Brownian motion. Computer modelling and new Technologies 18(10) 516 – 21
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Computer Modelling and Information Technologies
