First time to exit of a continuous It\^o process: general moment estimates and L1-convergence rate for discrete time approximations

We establish general moment estimates for the discrete and continuous exit times of a general It\^o process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the L1 norm with an order 1/2 with respect to the mesh size.