  Exponential integrability properties of numerical approximations are a key
tool for establishing positive rates of strong and numerically weak convergence
for a large class of nonlinear stochastic differential equations. It turns out
that well-known numerical approximation processes such as Euler-Maruyama
approximations, linear-implicit Euler approximations, and some tamed Euler
approximations from the literature rarely preserve exponential integrability
properties of the exact solution. The main contribution of this article is to
identify a class of stopped increment-tamed Euler approximations which preserve
exponential integrability properties of the exact solution under minor
additional assumptions on the involved functions.
