Strong convergence of the Euler-Maruyama approximation for SDEs with unbounded drift

In this work, we prove strong convergence on small time interval of order 1/2 - epsilon for arbitrarily small epsilon > 0 of the Euler-Maruyama approximation for additive Brownian motion with Holder continuous drift satisfying a linear growth condition. The proof is based on direct estimations of functional of the Euler-Maruyama approximation. The order of convergence does not depend on the Holder index of the drift, thus generalizing the results obtained in [10] to both Linear growth and to an optimal convergence order.