Estimation for change point of discretely observed ergodic diffusion processes

Abstract

We treat the change point problem in ergodic diffusion processes from discrete observations. Tonaki et al. ( 2020) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion models. When any changes are detected by this method, the next question to be considered is where the change point is. Therefore, we propose the method to estimate the change point of the parameter for two cases: the case where there is a change in the diffusion parameter, and the case where there is no change in the diffusion parameter but a change in the drift parameter. Furthermore, we present rates of convergence and distributional results of the change point estimators. Some examples and simulation results are also given.

Change point inference in ergodic diffusion processes based on high frequency data

“…In this paper, we consider the change point problem for ergodic diffusion processes. The change point problem for diffusion processes based on discrete observations has been studied by many researchers, for non-ergodic, see De Gregorio and Iacus (2008), Iacus and Yoshida (2012) and for ergodic, see Song and Lee (2009), Lee (2011), Negri and Nishiyama (2017), Song (2020), Tonaki et al (2020Tonaki et al ( ,2021 and reference therein. De Gregorio and Iacus (2008) considered the test for changes in the diffusion parameter and the change point estimation in non-ergodic case, and Iacus and Yoshida (2012) generalized the estimation method.…”

“…Song and Lee (2009), Lee (2011), andSong (2020) considered the test for changes in the diffusion parameter in ergodic case. Furthermore, Negri and Nishiyama (2017) and Tonaki et al (2020) proposed simultaneous test and adaptive tests for changes in the diffusion and drift parameters, respectively, and Tonaki et al (2021) treated change point estimation for the case where there is a change in the diffusion parameter and for the case where there is no change in the diffusion parameter but there is a change in the drift parameter. The method proposed by Tonaki et al (2020Tonaki et al ( ,2021 is summarized as follows.…”

“…The purposes of the change point problem in the ergodic diffusion processes are to investigate whether there is a change point in the diffusion and drift parameters, and if so, where it is. Given the data for the paths as shown in Figure 1, using the method of Tonaki et al (2020Tonaki et al ( ,2021, we can infer that the diffusion parameter changes at t = 150 and t = 350 for (A) and (D), respectively, and that there is no change in the diffusion parameter, but the drift parameter changes at t = 250 for (C). Furthermore, for (B), it can be inferred that there is no change in either the diffusion or drift parameters.…”

“…Furthermore, for (B), it can be inferred that there is no change in either the diffusion or drift parameters. However, Tonaki et al (2020Tonaki et al ( ,2021 do not provide discussion for the change of the drift parameter when there is a change in the diffusion parameter such as (A) and (D). In other words, we have no way to infer that there is no change point in the drift parameter in (A), and that the drift parameter changes at t = 200 in (D).…”

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We deal with the change point problem in ergodic diffusion processes based on high frequency data. Tonaki et al (2020Tonaki et al ( , 2021 studied the change point problem for the ergodic diffusion process model. However, the change point problem for the drift parameter when the diffusion parameter changes is still open.

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