Convergence in Total Variation of the Euler--Maruyama Scheme Applied to Diffusion Processes with Measurable Drift Coefficient and Additive Noise

We are interested in the Euler--Maruyama discretization of a stochastic differential equation dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In scheme, randomization time variable is used to get rid any regularity assumption this variable. prove weak convergence order 1/2 total variation distance. When has spatial divergence sense distributions $\rho$th power integrable respect Lebesgue measure space uniformly for some $\rho \ge d$, at terminal improves 1 up logarithmic factor. $d=1$, result preserved when derivative mass time. confirm our theoretical analysis by numerical experiments.