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. (2021a) proposed adaptive tests for detecting changes in the diffusion and drift parameters in ergodic diffusion process models. When any change in the diffusion or drift parameter is detected by this or any other method, the next question to consider is where the change point is located. 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. K E Y W O R D S adaptive tests, change point estimation, diffusion processes, discrete observations, test for parameter change 1

Parameter estimation for second-order SPDEs in multiple space dimensions

“…A crucial distinction that arises when transitioning from one spatial dimension to higher dimensions is that the random filed, denoted as X t (y), is not square integrable when considering a white noise, i.e., E[||X id t || 2 ϑ ] = ∞, where Q = id denotes the identity operator. Remarkably, this phenomenon persists even in two spatial dimensions, as demonstrated by the authors in [24]. To rectify this issue and ensure finite variance of the paths, it becomes necessary to employ a coloured cylindrical Wiener process instead of a white noise.…”

“…This methodology is widely adopted by researchers in the field, see, for instance [20], [18], or [15]. Moreover, we extend the two-dimensional approach presented by [24].…”

“…In our model, we incorporate a Q-Wiener process to account for the stochastic noise. By implementing damping mechanisms, [24] have successfully devised statistical inference techniques based on high-frequency observations, leveraging a spectral approach within the context of two spatial dimensions.…”

“…While power variations have received considerable attention in the context of one-dimensional SPDEs, their utilization for SPDEs in multiple spatial dimensions remains in its nascent stages. In a pivotal contribution, [24] analysed a two-dimensional SPDE model, laying the groundwork for further research in the realm of second-order, linear multi-dimensional SPDEs. In this endeavor, we follow a theoretical framework related to [24], adapting it to accommodate multiple spatial dimensions.…”

“…In a pivotal contribution, [24] analysed a two-dimensional SPDE model, laying the groundwork for further research in the realm of second-order, linear multi-dimensional SPDEs. In this endeavor, we follow a theoretical framework related to [24], adapting it to accommodate multiple spatial dimensions.…”