Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients

Abstract

Recently, explicit tamed schemes were proposed to approximate the SDEs with the non-Lipschitz continuous coefficients. This work proposes a semi-tamed Euler scheme, which is also explicit, to solve the SDEs with the drift coefficient equipped with the Lipschitz continuous part and non-Lipschitz continuous part. It is shown that the semi-tamed Euler converges strongly with the standard order one-half to the exact solution of the SDE. We also investigate the stability inheritance of the semi-tamed Euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution. Numerical experiments confirm the theoretical analysis.