Convergence analysis of an iterative algorithm to solve system of nonlinear stochastic Itô‐Volterra integral equations

Abstract

In this paper, an efficient and accurate numerical iterative algorithm based on the linear spline interpolation for solving the system of nonlinear stochastic Itô-Volterra integral equations is presented. The most important merit of this method is that it does not need to solve any system of nonlinear algebraic equations. An upper bound for the linear spline approximation of the stochastic function is provided. Using this upper bound and under the Lipschitz and linear growth conditions, the convergence analysis of the suggested method is studied.Finally, to verify the efficiency of the proposed scheme, some problems in the finance, physics, and biology are investigated, and the obtained results are compared with the stochastic -method. KEYWORDSlinear spline interpolation, multiple stock models, pendulum problem, predator-prey models, successive approximations method, system of stochastic Volterra integral equations MSC CLASSIFICATION 60H10; 60H20; 65C30Math Meth Appl Sci. 2020;43:5212-5233. wileyonlinelibrary.com/journal/mma

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“…18 In most cases, it is difficult or impossible to obtain an analytical solution for the integral equation (1). Therefore, in recent decades, there has been much attention to constructing some numerical techniques to achieve the approximate solutions with high accuracy for the VIDEs such as the homotopy perturbation method, 19 variational iteration method (VIM), 20,21 operational matrix method based on orthogonal polynomials 1,[22][23][24][25][26][27] operational tau method, 28,29 collocation approach, [30][31][32] block-pulse functions method, 33,34 radial basis function method, 3,35 wavelet method, 36,37 hat functions method, 38 least squares method, 10,39,40 hybrid functions method, [41][42][43][44] fixed point method, 45 successive approximations method, [46][47][48] Euler polynomials, 49,50 delta functions, 51 Taylor method, 52 and so on.…”

Numerical Solution of Multidimensional Stochastic Itô-Volterra Integral Equation Based on the Least Squares Method and Block Pulse Function

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