Wilson wavelets for solving nonlinear stochastic integral equations

A new computational method based on Wilson wavelets is proposed for solving a class of nonlinear stochastic It^{o}-Volterra integral equations. To do this a new stochastic operational matrix of It^{o} integration for Wilson wavelets is obtained. Block pulse functions (BPFs) and collocation method are used to generate a process to forming this matrix. Using these basis functions and their operational matrices of integration and stochastic integration, the problem under study is transformed to a  system of nonlinear algebraic equations which can be simply solved to obtain an approximate solution for the main problem. Moreover, a new technique for computing nonlinear terms in such problems is presented. Furthermore, convergence of Wilson wavelets expansion is investigated. Several examples are presented to show the efficiency and accuracy of the proposed method.