  The time at which a one-dimensional continuous strong Markov process attains
a boundary point of its state space is a discontinuous path functional and it
is, therefore, unclear whether the exit time can be approximated by hitting
times of approximations of the process. We prove a functional limit theorem for
approximating weakly both the paths of the Markov process and its exit times.
In contrast to the functional limit theorem in [arXiv:1902.06249v1] for
approximating the paths, we impose a stronger assumption here. This is
essential, as we present an example showing that the theorem extended with the
convergence of the exit times does not hold under the assumption in
[arXiv:1902.06249v1]. However, the EMCEL scheme introduced in
[arXiv:1902.06249v1] satisfies the assumption of our theorem, and hence we have
a scheme capable of approximating both the process and its exit times for every
one-dimensional continuous strong Markov process, even with irregular behavior
(e.g., a solution of an SDE with irregular coefficients or a Markov process
with sticky features). Moreover, our main result can be used to check for some
other schemes whether the exit times converge. As an application we verify that
the weak Euler scheme is capable of approximating the absorption time of the
CEV diffusion and that the scale-transformed weak Euler scheme for a squared
Bessel process is capable of approximating the time when the squared Bessel
process hits zero.
