Approximating exit times of continuous Markov processes

Thomas Kruse; Mikhail Urusov

doi:10.3934/dcdsb.2020076

Article Contents

2020, Volume 25, Issue 9: 3631-3650. Doi: 10.3934/dcdsb.2020076

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Approximating exit times of continuous Markov processes

Thomas Kruse^1, , and
Mikhail Urusov^2,

1.
Institute of Mathematics, University of Gieẞen, Arndtstr. 2, 35392 Gieẞen, Germany
2.
Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany

^* Corresponding author: Thomas Kruse
^* Corresponding author: Thomas Kruse

Received: September 2019

Revised: November 2019

Early access: January 2020

Published: September 2020

We acknowledge the support from the German Research Foundation through the project 415705084

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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 [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.

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Mathematics Subject Classification: Primary: 60J22, 60J25, 60J60; Secondary: 60H35, 60F17.

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