| Abstract: |
| As a well-known fact, the classical Euler scheme works merely for SDEs with coefficients of linear growth. In this paper, we propose a novel variant of Euler schemes, which is applicable to SDEs with super-linear drifts and encompasses the classical Euler scheme, the tamed Euler scheme and the truncated Euler scheme, as three typical candidates. On the one hand, by exploiting an approach based on the refined basic coupling, we show that the proposed Euler recursion is exponentially contractive under a mixed probability distance (i.e., the total variation distance plus the $L^1$-Wasserstein distance). On the other hand, by invoking a trick on the coupling by reflection, we, in particular, demonstrate that the tamed Euler scheme is exponentially contractive under the $L^1$-Wasserstein distance. In addition, as an important application, a quantitative $L^1$-Wasserstein error bound between the exact invariant probability measure and the numerical counterpart concerning the SDEs under consideration and the associated tamed Euler scheme, respectively. Last but not least, we want to stress that the theory derived in the present work is available for SDEs, where, most importantly, the drifts involved are allowed to be highly nonlinear. |
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