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

Time consistency of multi-period distortion measures

Dynamic risk measures play an important role for the acceptance or non-acceptance of risks in a bank portfolio. Dynamic consistency and weaker versions like conditional and sequential consistency guarantee that acceptability decisions remain consistent in time. An important set of static risk measures are so-called distortion measures. We extend these risk measures to a dynamic setting within the framework of the notions of consistency as above. As a prominent example, we present the Tail-Value-at-Risk (TVaR

Statistical estimation of multivariate Ornstein-Uhlenbeck processes and applications to co-integration

Abstract Ornstein-Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein-Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time. JEL Classifications: C13, C2