Haar Wavelet Method for Series Expansion of Fractional Wiener Integral

10.29252/mmr.5.2.229
Abstract
Introduction

The stochastic calculus plays an important role in the study of stochastic integral equations and stochastic differential equations. The fractional Brownian motion has many applications in different branches of sciences such as economics, physics and biology.

In many situations, the exact solution of these equations are not available or finding their exact solution is a very difficult process. Thus, finding an accurate and efficient numerical method for solving stochastic differential equations, and stochastic integral equations is important. Researchers have applied various numerical methods such as Dirichlet forms, Euler approximation, Skorohod integral, etc. In this paper, we used Haar wavelet functions for solving fractional Wiener integrals. Moreover, the error analysis of the proposed method is investigated.

Material and methods

In this scheme, first we present the properties of the Haar wavelet functions then an efficient method based on these functions is proposed to estimate the solution of fractional Wiener integral with Hurst parameter

Results and discussion

We solve two numerical examples by using present method to demonstrate the efficiency and simplicity of the present method. For different values of , mean of error and standard deviation of error are shown in the tables. The obtained results confirm that proposed method enables us to find reasonable approximate solutions.

Conclusion

The Haar wavelet is the simplest possible wavelet, so proposed method is easy to implement and it is a powerful mathematical tool to obtain the numerical solution of various kind of problems. ./files/site1/files/52/11.pdf
Keywords
Haar wavelet functions
Fractional wiener integrals
Brownian motion
1. Bertoin J., "Sur une intégrale pour les processus á α-variation borne", The Annals of Probability, 17(4) (1989) 1521-1535.## 2. Guerra J., Nualart D., "Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion", Stochastic Analysis and Applications, 26(5) (2008) 1053-1075. ## 3. Lisei H., Soós A., "Approximation of stochastic differential equations driven by fractional Brownian motion", Seminar on Stochastic Analysis, Random Fields and Applications, 59, (2008) 227-241. ## 4. Mishura Y., Shevchenko G., "The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion", Stochastic, 80(5) (2008) 489-511. ## 5. Russo F., Vallois P., "Forward, backward and symmetric stochastic integration", Probability Theory and Related Fields, 97 (1993) 403-421. ## 6. Cortes J., Joda L., Villafuerte L., "Mean square numerical solution of random differential equations: facts and possibilities", Computers and Mathematics with Applications, 53, (2007) 1098-1106. ## 7. Cortes J. C., Joda L., Villafuerte L., "Numerical solution of random differential equations: A mean square approach", Mathematical and Computer Modelling, 45 (2007) 757-765. ## 8. Jankovic S., Ilic D., "One linear analytic approximation for stochastic integro-differential equations", Acta Mathematica, Scientia , 30 (2010) 1073-1085. ## 9. Khodabin M., Maleknejad K., Rostami M., Nouri M., "Numerical solution of stochastic differential equations by second order Runge-Kutta methods(2011), Mathematical and Computer Modelling, 53 (2011) 1910-1920. ## 10. Kloeden P. E., Platen E., "Numerical solution of stochastic differential equations", Berlin, Applications of Mathematics, Springer-Verlag (1999). ## 11. Murge M., Pachpatte B., "Successive approximations for solutions of second order stochastic integro-differential equations of Itô type", Indian Journal of Pure and Applied Mathematics, 21(3) (1990) 260-274. ## 12. Saito Y., Mitsui T., "Simulation of stochastic differential equations", Annals of the Institute of Statistical Mathematics, 45 (1993) 419-432. ## 13. Heydari M. H., Hooshmandasl M. R., Maalek Ghaini F. M., Cattani C., "A computational method for solving stochastic Itô-Volterra integral equations based on stochastic operational matrix for generalized hat basis functions, Journal of Computational Physics, 270 (2014) 402-415. ## 14. Zeng C., Yang Q., Chen Y. Q., "Lyapunov techniques for stochastic differential equations driven by fractional Brownian motion", Hindawi Publishing Corporation Abstract and Applied Analysis (2014). ## 15. Guerra J., Nualart D., "Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion, Stochastic Analysis and Applications (2008) 26. ## 16. Dai W., Heyde C. C., "Itô formula with respect to fractional Brownian motion and its application, Journal of Applied Mathematics and Stochastic Analysis, 9 (1996) 439-448. ## 17. Lin S. J., "Stochastic analysis of fractional Brownian motions", Stochastics Stochastics, 55, (1995) 121-140. ## 18. Dung N. T., "Fractional stochastic differential equations with applications to finance", Journal of Mathematical Analysis and Applications, 397 (2013) 334-348. ## 19. Kolmogorov A. N., "Wienershe Spiralen und einige andere interessante Kurven im Hilbertschen Raum", Comptes Rendus (Doklady) de ĺAcademie des Sciences de ĺURSS, 26, (1940) 115-118. ## 20. Mandelbrot B. B., Van Ness J. W., "Fractional Brownian motions, fractional noises and applications", SIAM Review, 10(4) (1968) 422-437. ## 21. Biagini F., Hu Y., Øksendal B., Zhang T., "Stochastic calculus for fractional Brownian motion and applications, London, Springer (2008). ## 22. Strang G., "Wavelets and dilation equations", SIAM, 31 (1989) 614-627. ## 23. Lepik U., "Numerical solution of differential equations using Haar wavelets", Mathematics and Computers in Simulation, 68 (2005) 127-143. ## 24. Mohammadi F., "Haar wavelets approach for solving multidimensional stochastic Itô-Volterra integral equations", Applied Mathematics E-Notes, 15 (2015) 80-96. ## 
Mathematical Research
Volume 5, Issue 2 - Serial Number 11
Autumn 2019
Pages 229-236