On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.
Martin Hutzenthaler. Arnulf Jentzen. Peter E. Kloeden. "Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients." Ann. Appl. Probab. 22 (4) 1611 - 1641, August 2012. https://doi.org/10.1214/11-AAP803