A numerical scheme to solve nonlinear stochastic differential equation

Some traditional numerical methods to solve nonlinear stochastic differential equations (SDEs) have usually been established on Taylor series formulas. One of the deficiencies of these methods is that we should repeat these methods many times to obtain the various moments of the numerical solution of the SDE. To avoid these repetitions we have tried to prepare conditions that we can approximate the drift and the diffusion of the SDE using basic linear functions. By applying this approach we can obtain a sufficient continuous piecewise linear stochastic differential equation corresponding to the nonlinear SDE. Also we can approximate various moments of the solution of the original nonlinear SDE without any repetition. In addition, we prove that the solution of the obtained linear SDE is an approximated solution of the nonlinear SDE in the mean square sense. Finally, the results of the proposed method are compared with those of some efficient methods and the efficiency of the proposed method has been shown to be improved.