Abstract
The relationship between parabolic stochastic partial differential equations and autoregressive moving average (ARMA) time series on the real line is established. This is done in light of semigroup theory, under which the parabolic stochastic partial differential equation admits an Ornstein–Uhlenbeck process in Hilbert space. Hilbert-valued AR(1) (or, ARH(1) for short) processes are shown to naturally appear from sampled Ornstein–Uhlenbeck processes. An error representation of approximating AR(1) time series for evaluated ARH(1) processes is derived for the time dimension. Further, by proper projections of ARH(1) processes into Hilbertian subspaces a spatial error representation is derived for evaluation of such projections. The result shows convergence for approximating ARMA times series with increasing spatial dimensions. A numerical example demonstrates our theoretical results for the stochastic heat equation. The results provide a functional data analysis approach to ARH(1) processes.
Originalsprache | Englisch |
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Seiten (von - bis) | 55-80 |
Fachzeitschrift | Stochasitcs: An international Journal of Probability and Stochastic Processes |
Jahrgang | 97 |
Ausgabenummer | 1 |
DOIs | |
Publikationsstatus | Veröffentlicht - Nov. 2024 |
Österreichische Systematik der Wissenschaftszweige (ÖFOS)
- 101019 Stochastik