On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient
For researchers analyzing numerical methods for SDEs with irregular coefficients, this resolves a gap by showing the standard rate holds despite drift discontinuities.
The paper proves that the Euler-Maruyama scheme for scalar SDEs with discontinuous drift achieves an L_p-error rate of at least 1/2 for all p in [1,∞), matching the rate for SDEs with Lipschitz coefficients.
Recently a lot of effort has been invested to analyze the $L_p$-error of the Euler-Maruyama scheme in the case of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. For scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient so far only an $L_p$-error rate of at least $1/(2p)-$ has been proven. In the present paper we show that under the latter conditions on the coefficients of the SDE the Euler-Maruyama scheme in fact achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$ as in the case of SDEs with Lipschitz coefficients.