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

作者：Fasen Vicky^*

来源：Bernoulli, 2010, 16(1): 51-79.

DOI：10.3150/08-BEJ174

• 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

We consider a positive stationary generalized Ornstein-Uhlenbeck process
V-t = e(-xi t) (integral(1)(0)e(xi s-) d eta s + V-0) for t >= 0,
and the increments of the integrated generalized Ornstein-Uhlenbeck process I-k = integral(k)(k-1)root V-t-dL(t,) k is an element of N, where (xi(t), eta(t), L-t)(t >= 0) is a three-dimensional Levy 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, contiunous-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 is an element of N). Furthermore, we present a central limit result for (I-k)(k is an element of N). Regular variation and point process convergence play a crucial role in establishing the statistics of' (V-t)(t >= 0) and (I-k)(k is an element of N). The theory can be applied to the COGARCH(1, 1) and the Nelson diffusion model.

• 出版日期2010-2