We consider a positive stationary generalized Ornstein–Uhlenbeck process
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Vt=e−ξt(∫0te ξs−dηs+V0) for t≥0,
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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.