Numerical solution of stochastic Itô-Volterra integral equations based on Bernstein multi-scaling polynomials

Abstract

In this paper, first, Bernstein multi-scaling polynomials (BMSPs) and their properties are introduced. These polynomials are obtained by compressing Bernstein polynomials (BPs) under sub-intervals. Then, by using these polynomials, stochastic operational matrices of integration are generated. Moreover, by transforming the stochastic integral equation to a system of algebraic equations and solving this system using Newton’s method, the approximate solution of the stochastic Itô-Volterra integral equation is obtained. To illustrate the efficiency and accuracy of the proposed method, some examples are presented and the results are compared with other methods.