  In this article we propose a new explicit Euler-type approximation method for
stochastic differential equations (SDEs). In this method, Brownian increments
in the recursion of the Euler method are replaced by suitable bounded functions
of the Brownian increments. We prove strong convergence rate one-half for a
large class of SDEs with polynomial coefficient functions whose local
monotonicity constant grows at most like the logarithm of a Lyapunov-type
function.
