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}\biggl(\int_0^t\mathrm{e}^{\xi_{s-}}\ ,\mathrm{d}\eta_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 $(\xi_t,\eta_t,L_t)_{t\geq0}$ 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 $\operatorname {ARCH}(1)$ and $\operatorname {GARCH}(1,1)$ processes. In this paper we investigate the asymptotic behavior of extremes and the sample autocovariance function of $(V_t)_{t\geq0}$ 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 $(V_t)_{t\geq0}$ and $(I_k)_{k\in\mathbb{N}}$. The theory can be applied to the $\operatorname {COGARCH}(1,1)$ and the Nelson diffusion model.Comment: 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

Dependence Estimation for High Frequency Sampled Multivariate CARMA Models

The paper considers high frequency sampled multivariate continuous-time ARMA (MCARMA) models, and derives the asymptotic behavior of the sample autocovariance function to a normal random matrix. Moreover, we obtain the asymptotic behavior of the cross-covariances between different components of the model. We will see that the limit distribution of the sample autocovariance function has a similar structure in the continuous-time and in the discrete-time model. As special case we consider a CARMA (one-dimensional MCARMA) process. For a CARMA process we prove Bartlett's formula for the sample autocorrelation function. Bartlett's formula has the same form in both models, only the sums in the discrete-time model are exchanged by integrals in the continuous-time model. Finally, we present limit results for multivariate MA processes as well which are not known in this generality in the multivariate setting yet

Inclusive Art Education as a Tool for Art Museum Experiences

This Thesis looks at art education as a tool for meaningful art museum experiences. The study follows students on field trips at the High Art Museum as a process to develop a fully inclusive art educational program. The focus of the process of inclusivity evaluates the personal, socio-cultural, and physical aspects of the learning environment within the museum. The research questions that guide the study are 1) how the inclusive strategies I have found in my art classroom can be utilized to create an inclusive art museum setting, 2) how we can approach inclusive student learning an art museum environment, and 3) how the art museum field trip experience can be designed to create an engaging and worthwhile experience for students on the autism spectrum. This study offers a model of inclusion to an artistic learning environment

Extremal behavior of stochastic volatility models

Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility has - sometimes quite substantial - upwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by Levy driven volatility processes as, for instance, by Levy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Levy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,1) model. Driven by an arbitrary Levy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels