Document Type: Research Paper

Authors

Department of Mathematics, Khomein Branch, Islamic Azad University, Khomein, Iran

Abstract

This paper introduces a numerical method for solving the vasicek model by using a stochastic operational matrix based on the triangular functions (TFs) in combination with the collocation method. The method is stated by using conversion the vasicek model to a stochastic nonlinear system of $2m+2$ equations and $2m+2$ unknowns. Finally, the error analysis and some numerical examples are provided to demonstrate applicability and accuracy of this method.

Keywords

Main Subjects

[1] A. Deb, A. Dasgupta and G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems. J. Franklin Inst, vol. 343, (2006) 1-26.

[2] B. Oksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, SpringerVerlag, New York, 1998.

[3] E. Babolian, H. R. Marzban, and M. Salmani, Using triangular orthogonal functions for solving fredholm integral equations of the second kind, Appl. Math. Comput, vol. 201, (2008) 452-456.

[4] E. Babolian, Z. Masouri and S. Hatamzadeh-Varmazya, A direct method for numerically solving integral equations system using orthogonal triangular functions, Int. J. Industrial Mathematics, vol. 1, no. 2, (2009) 135-145, 2009.

[5] E. Pardoux and P. Protter, Stochastic volterra equations with anticipating coefficients, Ann. Probab, vol. 18, (1990) 1635-1655.

[6] K. Maleknejad, H. Almasieh and M. Roodaki, Triangular functions (TF) method for the solution of volterrafredholm integral equations, Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, (2009) 3293-3298.

[7] K. Maleknejad, and Z. Jafari Behbahani, Applications of two-dimensional triangular functions for solving nonlinear class of mixed volterra-fredholm integral equations, Mathematical and Computer Modelling.

[8] K. Maleknejad, M. Khodabin, and M. Rostami, A numerical method for solving m-dimensional stochastic Itvolterra integral equations by stochastic operational matrix, Computers and Mathematics with Applications, vol. 63, (2012) 133-143.

[9] K. Maleknejad, M. Khodabin and M. Rostami, Numerical solution of stochastic volterra integral equations by stochastic operational matrix based on block pulse functionsx, Mathematical and Computer Modelling, vol. 55, (2011) 791-800.

[10] M. A. Berger and V.J. Mizel, Volterra equations with Ito integrals I, J. Integral Equations vol. 2, no. 3, (1980) 187-245.

[11] M. Khodabin, K. Maleknejad and F. Hosseini, Application of triangular functions to numerical solution of stochastic volterra integral equations, IAENG International Journal of Applied Mathematics, 2013, IJAM- 43-1-01.

[12] P. E. Kloeden and E. Platen, Numerical solution of stochastic differential equations, Applications of Mathematics, Springer-Verlag, Berlin, 1999.