## Abstract and Applied Analysis

### Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation

#### Abstract

An efficient method to determine a numerical solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (FBM) with Hurst parameter $H\in (1/2,1)$ and $n$ independent one-dimensional standard Brownian motion (SBM) is proposed. The method is stated via a stochastic operational matrix based on the block pulse functions (BPFs). With using this approach, the SDE is reduced to a stochastic linear system of $m$ equations and $m$ unknowns. Then, the error analysis is demonstrated by some theorems and defnitions. Finally, the numerical examples demonstrate applicability and accuracy of this method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 523163, 11 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605903

Digital Object Identifier
doi:10.1155/2014/523163

Mathematical Reviews number (MathSciNet)
MR3182287

Zentralblatt MATH identifier
07022551

#### Citation

Ezzati, R.; Khodabin, M.; Sadati, Z. Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 523163, 11 pages. doi:10.1155/2014/523163. https://projecteuclid.org/euclid.aaa/1412605903

#### References

• J. Bertoin, “Sur une intégrale pour les processus à $\alpha$-variation bornée,” The Annals of Probability, vol. 17, no. 4, pp. 1277–1699, 1989.
• P. Cheng and M. Webster, “Stability analysis of impulsive stochastic functional differential equations with delayed impulses via comparison principle and impulsive delay differential inequality,” Abstract and Applied Analysis, vol. 2014, Article ID 710150, 2014.
• W. Gao, F. Deng, R. Zhang, and W. Liu, “Finite-time ${H}_{\infty }$ control for time-delayed stochastic systems with Markovian switching,” Abstract and Applied Analysis, vol. 2014, Article ID 809290, 2014.
• J. Guerra and D. Nualart, “Stochastic differential equations driven by fractional Brownian motion and standard Brownian motion,” Stochastic Analysis and Applications, vol. 26, no. 5, pp. 1053–1075, 2008.
• D. Huang and S. K. Nguang, “Robust ${H}_{\infty }$ static output feedback control of fuzzy systems: an ILMI approach,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 36, no. 1, pp. 216–222, 2006.
• A. Mark, F. Yao, and M. Hua, “Abstract functional stochastic evolution equations driven by fractional Brownian motion,” Abstract and Applied Analysis, vol. 2014, Article ID 516853, 2014.
• K. Maleknejad, M. Khodabin, and M. Rostami, “A numerical method for solving $m$-dimensional stochastic Itô-Volterra integral equations by stochastic operational matrix,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 133–143, 2012.
• K. Maleknejad, M. Khodabin, and M. Rostami, “Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 791–800, 2012.
• J. Wang and K. Zhang, “Non-fragile ${H}_{\infty }$ control for stochastic systems with Markovian jumping parameters and random packet losses,” Abstract and Applied Analysis, vol. 2014, Article ID 934134, 2014.
• H. Zhang, Y. Shi, and A. Saadat Mehr, “Robust static output feedback control and remote PID design for networked motor systems,” IEEE Transactions on Industrial Electronics, vol. 58, no. 12, pp. 5396–5405, 2011.
• M. Zähle, “Integration with respect to fractal functions and stochastic calculus. I,” Probability Theory and Related Fields, vol. 111, no. 3, pp. 333–374, 1998.
• L. Coutin, “An introduction to (stochastic) calculus with respect to fractional Brownian motion,” in Séminaire de Probabilités XL, vol. 1899, pp. 3–65, Springer, Berlin, Germany, 2007.
• L. Decreusefond and A. S. Üstünel, “Fractional Brownian motion: theory and applications,” in Systèmes Différentiels Fractionnaires, vol. 5 of ESAIM Proceedings, pp. 75–86, 1998.
• D. Nualart and A. Răşcanu, “Differential equations driven by fractional Brownian motion,” Collectanea Mathematica, vol. 53, no. 1, pp. 55–81, 2002.
• H. Lisei and A. Soós, “Approximation of stochastic differential equations driven by fractional Brownian motion,” in Seminar on Stochastic Analysis, Random Fields and Applications Progress in Probability, vol. 59, pp. 227–241, 2008.
• Y. Mishura and G. Shevchenko, “The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion,” Stochastics, vol. 80, no. 5, pp. 489–511, 2008.
• F. Russo and P. Vallois, “Forward, backward and symmetric stochastic integration,” Probability Theory and Related Fields, vol. 97, no. 3, pp. 403–421, 1993.
• F. Biagini, Y. Hu, B. ${\text{\O}}$ksendal, and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer, London, UK, 2008.
• T. Caraballo, M. J. Garrido-Atienza, and T. Taniguchi, “The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional Brownian motion,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 11, pp. 3671–3684, 2011.
• L. Longjin, F.-Y. Ren, and W.-Y. Qiu, “The application of fractional derivatives in stochastic models driven by fractional Brownian motion,” Physica A, vol. 389, no. 21, pp. 4809–4818, 2010.
• B. ${\text{\O}}$ksendal, Stochastic Differential Equations. An Introduction with Application, Springer, New York, NY, USA, 5th edition, 1998. \endinput