J. Math. Computer Sci., 31 (2023), 448–460 Online: ISSN 2008-949X Journal Homepage: www.isr-publications.com/jmcs Numerical approximation of p-dimensional stochastic Volterra integral equation using Walsh function Prit Pritam Paikaray, Sanghamitra Beuria ∗ , Nigam Chandra 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. Keywords: Stochastic volterra integral equation, Brownian motion, Itˆ o integral, Walsh approximation, Lipschitz condition. 2020 MSC: 60H05, 60H35, 65C30. ©2023 All rights reserved. 1. Introduction In recent decades, stochastic integral equations [13, 14] have been widely used in different fields like financial mathematics [3], physics, biology, engineering, and many others. Since it is not always possible to have an exact solution to the problem, numerical approximation to the integral equation becomes vital. To approximate the stochastic integral equation, orthogonal functions like block pulse function, Haar wavelet, Legendre polynomial, and others have been applied to approximate the stochastic integral equation [7–12, 16–18]. The fact that a computer can accurately estimate any Walsh function’s (which is a binary-valued function that takes values 1 and -1) current value at any given time gives it a significant edge over traditional trigonometric functions. Chen and Hsiao solved the variational problem using the Walsh function [1] in 1975. In 1979, they solved the integral equation using the same concept [6]. In this paper, we used the Walsh function to approximate the following p-dimensional stochastic Volterra integral equation (SVIE) x(t)= f(t)+  t 0 k(s, t)x(s)ds + p  γ=1  t 0 k γ (s, t)x(s)dB γ (s), s, t ∈ [0, T ), ∗ Corresponding author Email addresses: paikaraypritpritam@gmail.com (Prit Pritam Paikaray), sbeuria108@gmail.com (Sanghamitra Beuria), ncparida@gmail.com (Nigam Chandra Parida) doi: 10.22436/jmcs.031.04.07 Received: 2023-03-24 Revised: 2023-04-18 Accepted: 2023-05-10