Bernoulli

Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein–Uhlenbeck processes

Vicky Fasen

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Abstract

We consider a positive stationary generalized Ornstein–Uhlenbeck process

Vt=eξt(0teξsdηs+V0)  for 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 (ξt, ηt, Lt)t≥0 is a three-dimensional Lévy process independent of the starting random variable V0. 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 (Vt)t≥0 and (Ik)k∈ℕ. Furthermore, we present a central limit result for (Ik)k∈ℕ. Regular variation and point process convergence play a crucial role in establishing the statistics of (Vt)t≥0 and (Ik)k∈ℕ. The theory can be applied to the COGARCH(1, 1) and the Nelson diffusion model.

Article information

Source
Bernoulli Volume 16, Number 1 (2010), 51-79.

Dates
First available in Project Euclid: 12 February 2010

Permanent link to this document
http://projecteuclid.org/euclid.bj/1265984704

Digital Object Identifier
doi:10.3150/08-BEJ174

Mathematical Reviews number (MathSciNet)
MR2648750

Zentralblatt MATH identifier
05815964

Keywords
continuous-time GARCH process extreme value theory generalized Ornstein–Uhlenbeck process integrated generalized Ornstein–Uhlenbeck process mixing point process regular variation sample autocovariance function stochastic recurrence equation

Citation

Fasen, Vicky. Asymptotic results for sample autocovariance functions and extremes of integrated generalized Ornstein–Uhlenbeck processes. Bernoulli 16 (2010), no. 1, 51--79. doi:10.3150/08-BEJ174. http://projecteuclid.org/euclid.bj/1265984704.


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