Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme

We are interested in approximating a multidimensional hypoelliptic diffusion process (Xt)t[greater-or-equal, slanted]0 killed when it leaves a smooth domain D. When a discrete Euler scheme with time step h is used, we prove under a noncharacteristic boundary condition that the weak error is upper bounded by , generalizing the result obtained by Gobet in (Stoch. Proc. Appl. 87 (2000) 167) for the uniformly elliptic case. We also obtain a lower bound with the same rate , thus proving that the order of convergence is exactly 1/2. This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.