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A numerical method for solving \(m\)-dimensional stochastic itô-Volterra integral equations by stochastic operational matrix. (English) Zbl 1238.65007

Summary: The multidimensional Itô-Volterra integral equations arise in many problems such as an exponential population growth model with several independent white noise sources. In this paper, we obtain a stochastic operational matrix of block pulse functions on interval \([0,1)\) to solve \(m\)-dimensional stochastic Itô-Volterra integral equations. By using block pulse functions and their stochastic operational matrix of integration, m-dimensional stochastic Itô-Volterra integral equations can be reduced to a linear lower triangular system which can be directly solved by forward substitution. We prove that the rate of convergence is \(O(h)\). Furthermore, a 95% confidence interval of the errors’ mean is made, the results shows that the approximate solutions have a credible degree of accuracy.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H20 Stochastic integral equations
45R05 Random integral equations
65R20 Numerical methods for integral equations
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[1] Khodabin, M.; Maleknejad, K.; Rostami, M.; Nouri, M., Numerical solution of stochastic differential equations by second order Runge-Kutta methods, Mathematical and Computer Modelling, 53, 1910-1920 (2011) · Zbl 1219.65009
[3] Kloeden, P. E.; Platen, E., (Numerical Solution of Stochastic Differential Equations. Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (1999), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1216.60052
[4] Cortes, J. C.; Jodar, L.; Villafuerte, L., Numerical solution of random differential equations: a mean square approach, Mathematical and Computer Modelling, 45, 757-765 (2007) · Zbl 1140.65012
[5] Oksendal, B., Stochastic Differential Equations: An Introduction with Applications (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0897.60056
[7] Berger, M. A.; Mizel, V. J., Volterra equations with Ito integrals I, Journal of Integral Equations, 2, 187-245 (1980) · Zbl 0442.60064
[8] Zhang, X., Euler schemes and large deviations for stochastic Volterra equations with singular kernels, Journal of Differential Equations, 244, 2226-2250 (2008) · Zbl 1139.60329
[9] Jankovic, S.; Ilic, D., One linear analytic approximation for stochastic integro-differential equations, Acta Mathematica Scientia, 30, 1073-1085 (2010) · Zbl 1240.60153
[10] Zhang, X., Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, Journal of Functional Analysis, 258, 1361-1425 (2010) · Zbl 1189.60124
[11] Maleknejad, K.; Safdari, H.; Nouri, M., Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions, International Journal of Systems Science, 42, 195-199 (2011) · Zbl 1210.65204
[12] Maleknejad, K.; Mahmoudi, Y., Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-Pulse functions, Applied Mathematics and Computation, 149, 799-806 (2004) · Zbl 1038.65147
[13] Maleknejad, K.; Sohrabi, S.; Rostami, Y., Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Applied Mathematics and Computation, 188, 123-128 (2007) · Zbl 1114.65370
[14] Maleknejad, K.; Rahimi, B., Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind, Communications in Nonlinear Science and Numerical Simulation, 16, 2469-2477 (2011) · Zbl 1221.65338
[15] Jiang, Z. H.; Schaufelberger, W., Block Pulse Functions and Their Applications in Control Systems (1992), Springer-Verlag · Zbl 0771.93016
[16] Rao, G. Prasada, Piecewise Constant Orthogonal Functions and their Application to Systems and Control (1983), Springer: Springer Berlin · Zbl 0518.93003
[17] Higham, Desmond J., An algorithmic introduction to numerical simulation of stochastic differential equations, Society for Industrial and Applied Mathematics. Society for Industrial and Applied Mathematics, SIAM Review, 43, 3, 525-546 (2001) · Zbl 0979.65007
[18] Tudor, C.; Tudor, M., Approximation schemes for Ito-Volterra stochastic equations, Boletin Sociedad Matemática Mexicana, 3, 1, 73-85 (1995) · Zbl 0849.65102
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