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

Abstract

We consider a positive stationary generalized Ornstein–Uhlenbeck process $$V_t=e^{−ξ_t} \left( ∫_0^te^{ ξ_{s−}}dη_s+V_0 \right) \mathrm{for}\quad t≥0,$$ 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.