Numerical Solution of Vasicek Equation by Using Brownian Wavelets and Multiple Ito-Integral

In this paper‎, ‎we present a new approach to solving stochastic differential equations and the Vasicek equation by using Brownian wavelets and multiple Ito-integral‎. ‎Firstly‎, ‎the calculation of the multiple Ito-integral based on the structure of Brownian motion is presented and the error of Ito-integrate computation is minimized under this condition‎. ‎Then‎, ‎the Brownian wavelets 1D and 3D based on coefficients Brownian motion are introduced‎. ‎After that‎, ‎a system of linear and nonlinear equations of coefficients Brownian motion is obtained such that by solving this system the approximate solution of the Vasicek equation is obtained‎. In the last section, ‎some numerical examples are given.