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Limit theorems for \(p\)-variations of solutions of SDEs driven by additive stable Lévy noise and model selection for Paleo-climatic data. (English) Zbl 1218.60027

Duan, Jinqiao (ed.) et al., Recent development in stochastic dynamics and stochastic analysis. Dedicated to Zhi-Yuan Zhang on the occasion of his 75th birthday. Hackensack, NJ: World Scientific (ISBN 978-981-4277-25-9/hbk). Interdisciplinary Mathematical Sciences 8, 161-175 (2010).
The article proves functional limit theorems for the \(p\)-variations of stochastic processes of the form \(X = Y + L\) where \(L\) is an \(\alpha\)-stable Lévy process, and \(Y\) is a process that satisfies some mild Lipschitz condition. In case \(X\) is the solution of a stochastic differential equation (SDE) driven by the process \(L\), this result can be used to estimate the stability index \(\alpha\). These results are then applied to paleo-climatic temperature time series taken from the Greenland ice core to estimate the parameter \(\alpha\) in a model which describes the time series as the solution of an SDE with \(\alpha\)-stable noise component.
Let \(X\) be a stochastic process. Define \[ V^n_p(X)_t := \sum_{i=1}^{[nt]} |\Delta^n_i X|^p \] The main result states that, for \(X = Y + L\), where \(L\) is an \(\alpha\)-stable Lévy process and \(Y\) satisfies appropriate conditions (which guarantee that \(Y\) will not contribute to the \(p\)-variation of \(X\)), and for \(p > \alpha/2\),
\[ (V^n_p(X)_t - ntB_n(\alpha, p))_{t \geq 0} \rightarrow (L^{'}_t)_{t\geq 0} \]
as \(n \rightarrow \infty\). Here, \(B_n(\alpha, p)\) is a suitable normalizing sequence, and \(L^{'}\) is an \(\alpha/p\)-stable Lévy process. The convergence is in distribution on Skorokhod space.
For the entire collection see [Zbl 1191.60005].

MSC:

60F17 Functional limit theorems; invariance principles
60G52 Stable stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
62F10 Point estimation
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
86A40 Glaciology
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