About solving stochastic It\^{o}-Volterra integral equations using the spectral collocation method

The purpose of this paper is to propose the spectral collocation method to solve linear and nonlinear stochastic It^o-Volterra integral equations (SVIEs). The proposed approach is different from other numerical techniques as we consider the Legendre Gauss type quadrature for estimating It^o integrals. The main characteristic of the presented method is that it reduces SVIEs into a system of algebraic equations. Thus, we can solve the problem by Newton's method. Furthermore,  the convergence analysis of the approach is established. The method is computationally attractive, and to reveal the accuracy, validity, and efficiency of the proposed method, some numerical examples and convergence analysis are included.

کلیدواژه ها:

stochastic ، It^o-Volterra integral equations ، collocation ، shifted Legendre polynomials ، Gauss type quadrature