  The Euler scheme is one of the standard schemes to obtain numerical
approximations of stochastic differential equations (SDEs). Its convergence
properties are well-known in the case of globally Lipschitz continuous
coefficients. However, in many situations, relevant systems do not show a
smooth behavior, which results in SDE models with discontinuous drift
coefficient. In this work, we will analyze the long time properties of the
Euler scheme applied to SDEs with a piecewise constant drift and a constant
diffusion coefficient and carry out intensive numerical tests for its
convergence properties. We will emphasize on numerical convergence rates and
analyze how they depend on properties of the drift coefficient and the initial
value. We will also give theoretical interpretations of some of the arising
phenomena. For application purposes, we will study a rank-based stock market
model describing the evolution of the capital distribution within the market
and provide theoretical as well as numerical results on the long time ranking
behavior.
