Numerical approximation of \(p\)dimensional stochastic Volterra integral equation using Walsh function
Authors
P. P. Paikaray
 Department of Mathematics, College of Basic Science and Humanities, OUAT, Bhubaneswar, Odisha, 751003, India.
S. Beuria
 Department of Mathematics, College of Basic Science and Humanities, OUAT, Bhubaneswar, Odisha, 751003, India.
N. Ch. Parida
 Department of Mathematics, College of Basic Science and Humanities, OUAT, Bhubaneswar, Odisha, 751003, India.
Abstract
In this paper, we propose a numerical approach for solving \(p\)dimensional stochastic Volterra integral equations using the Walsh function approximation. The main goal is to transform integral equations into an algebraic system and solve this further to get an approximate solution to the integral equation. The convergence and error analysis of the proposed method are studied for integral equations having functions in the Lipschitz class. The computation of various examples for which analytical solutions are available shows that the proposed method is more accurate than the existing techniques for solving linear \(p\)dimensional stochastic Volterra integral equations.
Share and Cite
ISRP Style
P. P. Paikaray, S. Beuria, N. Ch. Parida, Numerical approximation of \(p\)dimensional stochastic Volterra integral equation using Walsh function, Journal of Mathematics and Computer Science, 31 (2023), no. 4, 448460
AMA Style
Paikaray P. P., Beuria S., Parida N. Ch., Numerical approximation of \(p\)dimensional stochastic Volterra integral equation using Walsh function. J Math Comput SCIJM. (2023); 31(4):448460
Chicago/Turabian Style
Paikaray, P. P., Beuria, S., Parida, N. Ch.. "Numerical approximation of \(p\)dimensional stochastic Volterra integral equation using Walsh function." Journal of Mathematics and Computer Science, 31, no. 4 (2023): 448460
Keywords
 Stochastic volterra integral equation
 Brownian motion
 Itô integral
 Walsh approximation
 Lipschitz condition
MSC
References

[1]
C. F. Chen, C. H. Hsiao, A Walsh Series Direct Method for Solving Variational Problems, J. Franklin Inst., 300 (1975), 265–280

[2]
C. F. Cheng, Y. T. Tsay, T. T. Wu, Walsh operational matrices for fractional calculus and their application to distributed systems, J. Franklin Inst., 303 (1977), 267–284

[3]
A. Etheridge, A Course in Financial Calculus, Cambridge University Press, Cambridge (2002)

[4]
B. Golubov, A. Efimov, V. Skvortsov, Walsh series and transforms, Kluwer Academic Publishers Group, Dordrecht (1991)

[5]
S. HatamzadehVarmazyar, Z. Masouri, E. Babolian, Numerical method for solving arbitrary linear differential equations using a set of orthogonal basis functions and operational matrix, Appl. Math. Model., 40 (2016), 233–253

[6]
C. H. Hsiao, C. F. Chen, Solving integral equation via Walsh functions, Comput. Electr. Engrg., 6 (1979), 279–292

[7]
K. Maleknejad, B. Basirat, E. Hashemizadeh, Hybrid Legendre polynomials and blockpulse functions approach for nonlinear VolterraFredholm integrodifferential equations, Comput. Math. Appl., 61 (2011), 2821–2828

[8]
K. Maleknejad, M. Khodabin, M. Rostami, A numerical method for solving mdimensional stochastic ItˆoVolterra integral equations by stochastic operational matrix, Comput. Math. Appl., 63 (2012), 133–143

[9]
K. Maleknejad, M. Khodabin, M. Rostami, Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions, Math. Comput. Modelling, 55 (2012), 791–800

[10]
K. Maleknejad, S. Sohrabi, Y. Rostami, Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Appl. Math. Comput., 188 (2007), 123–128

[11]
F. Mohammadi, Haar wavelets approach for solving multidimensional stochastic ItˆoVolterra integral equations, Appl. Math. ENotes, 15 (2015), 80–96

[12]
F. Mohammadi, Numerical Solution of Stochastic ItoVolterra Integral Equations using Haar Wavelets, Numer. Math. Theory Methods Appl., 9 (2016), 416–431

[13]
P. E. Kloeden, E. Platen, Numerical solution of stochastic differential equations, SpringerVerlag, Berlin (1992)

[14]
B. Oksendal, Stochastic Differential Equations: An Introduction with Applications, SpringerVerlag, New York (1998)

[15]
G. P. Rao, Piecewise Constant Orthogonal Functions and Their Application to Systems and Control, SpringerVerlag, Berlin (1983)

[16]
S. Saha Ray, S. Singh, New stochastic operational matrix method for solving stochastic ItˆoVolterra integral equations characterized by fractional Brownian motion, Stoch. Anal. Appl., 39 (2021), 224–234

[17]
S. Singh, S. Saha Ray, Stochastic operational matrix of Chebyshev wavelets for solving multidimensional stochastic Itˆo Volterra integral equations, Int. J. Wavelets Multiresolut. Inf. Process., 17 (2019), 16 pages

[18]
A. A. C. Tofigha, M. Khodabina, R. Ezzatia, Numerical Solution of Linear Stochastic Volterra Integral Equations via New Basis Functions, Filomat, 33 (2019), 5959–5966

[19]
J. L. Walsh, A Closed Set of Normal Orthogonal Functions, Amer. J. Math., 45 (1923), 5–24