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

Abstract

We consider a positive stationary generalized Ornstein--Uhlenbeck process \[V_t=\mathrm{e}^{-ξ_t}\biggl(\int_0^t\mathrm{e}^{ξ_{s-}}\ ,\mathrm{d}η_s+V_0\biggr)\qquadfor t\geq0,\] and the increments of the integrated generalized Ornstein--Uhlenbeck process $I_k={\int }_{k-1}^k\sqrt{V_{t-}}\mathrm{d}{L}_t$, $k\in \mathbb{N}$, where $({\xi }_t,{\eta }_t,L_t{)}_{t\ge 0}$ 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 $\mathrm{ARCH}(1)$ and $\mathrm{GARCH}(1,1)$ processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t{)}_{t\ge 0}$ and $(I_k{)}_{k\in \mathbb{N}}$. Furthermore, we present a central limit result for $(I_k{)}_{k\in \mathbb{N}}$. Regular variation and point process convergence play a crucial role in establishing the statistics of $(V_t{)}_{t\ge 0}$ and $(I_k{)}_{k\in \mathbb{N}}$. The theory can be applied to the $\mathrm{COGARCH}(1,1)$ and the Nelson diffusion model.

Published in at http://dx.doi.org/10.3150/08-BEJ174 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (2)

• GARCH modelling in continuous time for irregularly spaced time series data
• Extremal behavior of stochastic integrals driven by regularly varying Lévy processes