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February 2010 Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein–Uhlenbeck processes
Vicky Fasen
Bernoulli 16(1): 51-79 (February 2010). DOI: 10.3150/08-BEJ174

Abstract

We consider a positive stationary generalized Ornstein–Uhlenbeck process $$V_t=e^{−ξ_t} \left( ∫_0^te^{ ξ_{s−}}dη_s+V_0 \right) \mathrm{for}\quad t≥0,$$ and the increments of the integrated generalized Ornstein–Uhlenbeck process $I_{k}=\int_{k-1}^{k}\sqrt{V_{t-}}\,\mathrm{d}L_{t}$, $k∈ℕ$, where $(ξ_t, η_t, L_t)_{t≥0}$ is a three-dimensional Lévy process independent of the starting random variable $V_0$. The genOU model is a continuous-time version of a stochastic recurrence equation. Hence, our models include, in particular, continuous-time versions of ARCH(1) and GARCH(1, 1) processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t)_{t≥0}$ and $(I_k)_{k∈ℕ}$. Furthermore, we present a central limit result for $(I_k)_{k∈ℕ}$. Regular variation and point process convergence play a crucial role in establishing the statistics of $(V_t)_{t≥0}$ and $(I_k)_{k∈ℕ}$. The theory can be applied to the COGARCH$(1, 1)$ and the Nelson diffusion model.

Citation

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Vicky Fasen. "Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein–Uhlenbeck processes." Bernoulli 16 (1) 51 - 79, February 2010. https://doi.org/10.3150/08-BEJ174

Information

Published: February 2010
First available in Project Euclid: 12 February 2010

zbMATH: 05815964
MathSciNet: MR2648750
Digital Object Identifier: 10.3150/08-BEJ174

Rights: Copyright © 2010 Bernoulli Society for Mathematical Statistics and Probability

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Vol.16 • No. 1 • February 2010
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