# Wasserstein convergence rates for random bit approximations of continuous Markov processes

## TL;DR

This paper analyzes the convergence rates of a coin-tossing Markov chain scheme for approximating one-dimensional continuous Markov processes, showing a rate of 1/4 in Wasserstein distance under mild conditions.

## Contribution

It establishes the convergence speed of a novel approximation scheme for continuous Markov processes using random bit-based Markov chains, including irregular cases.

## Key findings

- Convergence rate of 1/4 in Wasserstein distance for fixed times.
- Any path convergence rate is strictly less than 1/4.
- Applicable to processes with irregular behavior like SDEs with irregular coefficients.

## Abstract

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