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.