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

International audienceWe are interested in approximating a multidimensional hypoelliptic diffusion process $(X_t)_{t\geq 0}$ killed when it leaves a smooth domain $D$. When a discrete Euler scheme with time step $h$ is used, we prove under a non characteristic boundary condition that the weak error is upper bounded by $C_1\sqrt h$, generalizing the result obtained by the first author in Gobet'00 for the uniformly elliptic case. We also obtain a lower bound with the same rate $\sqrt h$, thus proving that the order of convergence is exactly $\frac 12$. This provides a theoretical explanation of the well-known bias that we can numerically observe in that kind of procedure