Higher order numerical methods for SDEs without globally monotone coefficients

In the present work, we delve into further study of numerical approximations of SDEs with non-globally monotone coefficients. We design and analyze a new family of stopped increment-tamed time discretization schemes of Euler, Milstein and order 1.5 type for such SDEs. By formulating a novel unified framework, the proposed methods are shown to possess the exponential integrability properties, which are crucial to recovering convergence rates in the non-global monotone setting. Armed with such exponential integrability properties and by the arguments of perturbation estimates, we successfully identify the optimal strong convergence rates of the aforementioned methods in the non-global monotone setting. Numerical experiments are finally presented to corroborate the theoretical results.