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

We are interested in approximating a multidimensional hypoelliptic diffusion process (Xt)t⩾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 C1h, 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 h, thus proving that the order of convergence is exactly 1/2. rovides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure.