# Approximating exit times of continuous Markov processes

## TL;DR

This paper establishes a functional limit theorem for approximating the exit times of one-dimensional continuous strong Markov processes, demonstrating the effectiveness of the EMCEL scheme and other schemes in capturing boundary hitting times even with irregular behaviors.

## Contribution

It provides a stronger convergence theorem for both paths and exit times of Markov processes, extending previous results and verifying the applicability of the EMCEL and weak Euler schemes.

## Key findings

- EMCEL scheme satisfies the theorem's assumptions for approximating exit times.
- Weak Euler scheme can approximate absorption times of CEV diffusion.
- Scale-transformed weak Euler scheme approximates hitting zero for squared Bessel process.

## Abstract

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.