Recently a lot of effort has been invested in the literature to analyze the Lp-error of the Euler 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 Lp-error rate of at least 1/(2p)− has been proven in the literature. In this talk we show that under the latter assumptions on the coefficients of the SDE the Euler scheme in fact achieves an Lp-error rate of at least 1/2 for all p ∈ [1, ∞) as in the case of SDEs with Lipschitz coefficients.
We furthermore present a numerical method based on finitely many evaluations of the driving Brownian motion, which achieves an Lp-error rate of at least 3/4 for all p ∈ [1, ∞) if, additionally to the assumptions stated above, both the drift and the diffusion coefficients are piecewise differentiable with Lipschitz derivatives. Finally, we show that the Lp-error rate 3/4 can not be improved in general by a numerical method based on finitely many evaluations of the driving Brownian motion.
The talk is based on joint work with Thomas Müller-Gronbach (University of Passau) and Arnulf Jentzen (University of Münster).