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 [3] 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 [3]. However, the EMCEL scheme introduced in [3] 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.

中文翻译：

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连续马尔可夫过程的近似出口时间

一维连续强马尔可夫过程到达其状态空间的边界点的时间是不连续的路径函数，因此尚不清楚是否可以通过命中过程的近似时间来近似退出时间。我们证明了一个函数极限定理，用于弱近似马尔可夫过程的路径及其退出时间。与[中的函数极限定理相反3为了近似路径，我们在这里强加一个假设。这一点很重要，因为我们举一个例子说明，随着出口时间收敛而扩展的定理在[3]。但是，[3]满足我们的定理为前提，因此，我们具有能够近似的进程和其退出时间的方案的每个一维连续强马尔可夫过程，甚至具有不规则的行为（例如，SDE的不规则系数或溶液具有粘性的马尔可夫过程）。此外，我们的主要结果可用于检查其他一些方案，退出时间是否收敛。作为一个应用，我们验证了弱Euler方案能够近似CEV扩散的吸收时间，并且平方贝塞尔过程的标度变换后的弱Euler方案能够近似于平方贝塞尔过程达到零的时间。