Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The is based on construction certain chains whose laws can be embedded into process with sequence stopping times. Under mild condition process' measure we prove that converge at fixed times rate 1/4 respect to every p-th Wasserstein distance. For paths, any strictly smaller than 1/4. Our results apply, in particular, processes irregular behavior such as solutions SDEs coefficients and sticky points.