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We determine the convergence speed of a numerical scheme for approximatingone-dimensional continuous strong Markov processes. The scheme is based on theconstruction of coin tossing Markov chains whose laws can be embedded into theprocess with a sequence of stopping times. Under a mild condition on theprocess' speed measure we prove that the approximating Markov chains convergeat fixed times at the rate of $1/4$ with respect to every $p$-th Wassersteindistance. For the convergence of paths, we prove any rate strictly smaller than$1/4$. Our results apply, in particular, to processes with irregular behaviorsuch as solutions of SDEs with irregular coefficients and processes with stickypoints.

Wasserstein convergence rates for random bit approximations of continuous Markov processes