Convergence rates of the Semi-Discrete method for stochastic differential equations

We study the convergence rates of the semi-discrete (SD) method originally proposed in Halidias (2012), Semi-discrete approximations for stochastic differential equations and applications, International Journal of Computer Mathematics, 89(6). The SD numerical method was originally designed mainly to reproduce qualitative properties of nonlinear stochastic differential equations (SDEs). The strong convergence property of the SD method has been proved, but except for certain classes of SDEs, the order of the method was not studied. We study the order of ${\mathcal L}^2$-convergence and show that it can be arbitrarily close to $1/2.$ The theoretical findings are supported by numerical experiments.