  Many stochastic differential equations (SDEs) in the literature have a
superlinearly growing nonlinearity in their drift or diffusion coefficient.
Unfortunately, moments of the computationally efficient Euler-Maruyama
approximation method diverge for these SDEs in finite time. This article
develops a general theory based on rare events for studying integrability
properties such as moment bounds for discrete-time stochastic processes. Using
this approach, we establish moment bounds for fully and partially
drift-implicit Euler methods and for a class of new explicit approximation
methods which require only a few more arithmetical operations than the
Euler-Maruyama method. These moment bounds are then used to prove strong
convergence of the proposed schemes. Finally, we illustrate our results for
several SDEs from finance, physics, biology and chemistry.
