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

We consider a positive stationary generalized Ornstein–Uhlenbeck process $$V_t={\mathrm{e}}^{-{\xi }_t}({\int }_0^t{\mathrm{e}}^{{\xi }_{s-}},\mathrm{d}{\eta }_s+V_0)\text{qquadfor}t\ge 0,$$ and the increments of the integrated generalized Ornstein–Uhlenbeck process $I_k={\int }_{k-1}^k\sqrt{V_{t-}}\mathrm{d}{L}_t$, $k\in \mathbb{N}$, where $({\xi }_t,{\eta }_t,L_t{)}_{t\ge 0}$ is a three-dimensional L'{e}vy 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.