On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient (Q2179625)

The authors consider the autonomous differential equation \[ \begin{aligned} dX_t & = \mu(X_t)dt + \sigma(X_t)dW_t, \quad t \in [0,1] \\ X_0 & = x_0 \end{aligned} \] They study the performance of the Euler scheme in the case where the drift coefficient has infinitely many discontinuities. They show the following estimate \[ (\mathbb{E}[\|X-X_n\|_q^p])^{1/p} \leq \begin{cases} c/ \sqrt{n}, & \text{ if } q < \infty,\\ c \sqrt{\ln (n+1)}/\sqrt{n}, & \text{ if } q = \infty. \end{cases} \]