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

by Th. Müller-Gronbach and L. Yaroslavtseva

(Report number 2018-50)

Abstract
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.

Keywords:

@Techreport{MY18_804,
  author = {Th. M\"uller-Gronbach and L. Yaroslavtseva},
  title = {On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-50},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-50.pdf },
  year = {2018}
}

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