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We establish general moment estimates for the discrete and continuous exittimes of a general It\^o process in terms of the distance to the boundary.These estimates serve as intermediate steps to obtain strong convergenceresults for the approximation of a continuous exit time by a discretecounterpart, computed on a grid. In particular, we prove that the discrete exittime of the Euler scheme of a diffusion converges in the L1 norm with an order1/2 with respect to the mesh size.

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