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Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions. (English) Zbl 1255.65247
Summary: We propose an efficient method for solving stochastic Volterra integral equations. By using block pulse functions and their stochastic operational matrix of integration, a stochastic Volterra integral equation can be reduced to a linear lower triangular system, which can be directly solved by forward substitution. The results show that the approximate solutions have a good degree of accuracy.

65R20 Numerical methods for integral equations
60H20 Stochastic integral equations
45R05 Random integral equations
Full Text: DOI
[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
[2] M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Interpolation solution in generalized stochastic exponential population growth model, Applied Mathematical Modelling, Manuscript (2011) (in press), Corrected Proof, Available online 23 July 2011. · Zbl 1243.60056
[3] Kloeden, P.E.; Platen, E., ()
[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] Cortes, J.C.; Jodar, L.; Villafuerte, L., Mean square numerical solution of random differential equations: facts and possibilities, Computers and mathematics with applications, 53, 1098-1106, (2007) · Zbl 1127.65003
[6] Oksendal, B., ()
[7] Holden, H.; Oksendal, B.; Uboe, J.; Zhang, T., Stochastic partial differential equations, (2009), Springer-Verlag New York
[8] Berger, M.A.; Mizel, V.J., Volterra equations with ito integrals I, Journal of integral equations, 2, 187-245, (1980) · Zbl 0442.60064
[9] Murge, M.G.; Pachpatte, B.G., On second order ito type stochastic integrodifferential equations, Analele stiintifice ale universitatii. I. cuzadin iasi, Mathematica, 25-34, (1984), Tomul xxx, s.I a · Zbl 0573.60048
[10] Murge, M.G.; Pachpatte, B.G., Succesive approximations for solutions of second order stochastic integrodifferential equations of ito type, Indian journal of pure and applied mathematics, 21, 3, 260-274, (1990) · Zbl 0703.60062
[11] 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
[12] Jankovic, S.; Ilic, D., One linear analytic approximation for stochastic integro-differential eauations, Acta Mathematica scientia, 30B, 4, 1073-1085, (2010) · Zbl 1240.60153
[13] Zhang, X., Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, Acta journal of functional analysis, 258, 1361-1425, (2010) · Zbl 1189.60124
[14] Yong, J., Backward stochastic Volterra integral equations and some related problems, Stochastic processes and their applications, 116, 779-795, (2006) · Zbl 1093.60042
[15] Maleknejad, K.; Tavassoli Kajani, M., Solving second kind integral equations by Galerkin methods with hybrid Legendre and block-pulse functions, Applied mathematics and computation, 145, 623-629, (2003) · Zbl 1101.65323
[16] 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
[17] 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
[18] Maleknejad, K.; Shahrezaee, M.; Khatami, H., Numerical solution of integral equations system of the second kind by block pulse functions, Applied mathematics and computation, 166, 15-24, (2005) · Zbl 1073.65149
[19] 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
[20] Babolian, E.; Maleknejad, K.; Mordad, M.; Rahimi, B., A numerical method to solve Fredholm-Volterra integral equations in two dimensional spaces using block pulse functions and operational matrix, Journal of computational and applied mathematics, 235, 14, 3965-3971, (2011) · Zbl 1219.65158
[21] Maleknejad, K.; Mahdiani, K., Solving nonlinear mixed volterra – fredholm integral equations with two dimensional block-pulse functions using direct method, Communications in nonlinear science and numerical simulation, (2011) · Zbl 1222.65149
[22] Maleknejad, K.; Basirat, B.; Hashemizadeh, E., Hybrid Legendre polynomials and block-pulse functions approach for nonlinear volterra – fredholm integro-differential equations, Computers & mathematics with applications, 61, 9, 2821-2828, (2011) · Zbl 1221.65333
[23] Maleknejad, K.; Hashemizadeh, E.; Basirat, B., Computational method based on bernestein operational matrices for nonlinear volterra – fredholm-Hammerstein integral equations, Communications in nonlinear science and numerical simulation, 17, 1, 52-61, (2012) · Zbl 1244.65243
[24] Jiang, Z.H.; Schaufelberger, W., Block pulse functions and their applications in control systems, (1992), Springer-Verlag · Zbl 0771.93016
[25] Prasada Rao, G., Piecewise constant orthogonal functions and their application to systems and control, (1983), Springer Berlin · Zbl 0518.93003
[26] Tudor, C.; Tudor, M., Approximation schemes for ito – volterra stochastic equations, Boletin sociedad matemática mexicana, 3, 1, 73-85, (1995) · Zbl 0849.65102
[27] Etheridge, A., A course in financial calculus, (2002), Cambridge University Press · Zbl 1002.91025
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