The projected explicit Itô–Taylor methods for stochastic differential equations under locally Lipschitz conditions and polynomial growth conditions

Although the numerical methods to stochastic differential equations with the coefficients of locally Lipschitz and polynomial growth have been discussed commonly by some authors, there are few works on the high strong order numerical methods. In this paper, the mean-square convergence of the general projected explicit Itô–Taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition, suitable local Lipschitz conditions and polynomial growth conditions. Our analysis follows the idea of stochastic C-stability and stochastic B-consistency. After giving the selection strategy of optimal parameters, we prove that the projected explicit Itô–Taylor methods, owning optimal parameters, share the same mean-square convergence orders with non-projected ones. Finally, two numerical experiments are presented to show the effectiveness of theoretical results.