Euler polynomial solutions of nonlinear stochastic Itô-Volterra integral equations

In this study, a practical matrix method based on operational matrices of integration and collocation points is presented to find the approximate solution of nonlinear stochastic Itô-Volterra integral equations. For this aim, first we compute the stochastic operational matrix of Euler polynomials. Then by using this operational matrix, nonlinear stochastic Itô-Volterra integral equation is reduced to a system of nonlinear algebraic equations with unknown Euler coefficients which can be solved by using a convenient numerical method such as Newton's method. Also, convergence analysis and error estimation of the approach is discussed and an upper error bound is provided under mild conditions. Finally, some numerical examples are provided to confirm the accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC by running some programs written in MATLAB software.