In this thesis 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.