7.2.4 Higher order integration schemes

We can obtain more accurate Taylor schemes by including further multiple stochastic integrals from the stochastic Taylor expansions () and () [40]. However, their development and implementation is very tedious and in general the improvement in accuracy is not needed. This is so, because one has to solve numerically the Langevin equations and average the results for different realizations of the noise. This generates a source of statistical errors which are in many occasions greater than the systematic errors due to the order of convergence of the numerical method. So it is usually better to spend the computer time in reducing the statistical errors by increasing the number of samples in the average rather than using a more complicated higher order algorithm [45].