Risk contagion concerns any entity dealing with large scale risks. Suppose Z = (Z1, Z2) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of Z2 on Z1. This can be measured in a variety of ways. In this paper, we study two such measures: the quantity E[(Z1 − t)+|Z2 > t] called Marginal Mean Excess (MME) as well as the related quantity E[Z1|Z2 > t] called Marginal Expected Shortfall (MES). Both quantities are indicators of risk contagion and useful in various applications ranging from finance, insurance and systemic risk to environmental and climate risk. We work under the assumptions of multivariate regular variation, hidden regular variation and asymptotic tail independence for the risk vector Z. Many broad and useful model classes satisfy these assumptions. We present several examples and derive the asymptotic behavior of both MME and MES as the threshold t → ∞. We observe that although we assume asymptotic tail independence in the models, MME and MES converge to ∞ under very general conditions; this reflects that the underlying weak dependence in the model still remains significant. Besides the consistency of the empirical estimators we introduce an extrapolation method based on extreme value theory to estimate both MME and MES for high thresholds t where little data are available. We show that these estimators are consistent and illustrate our methodology in both simulated and real data sets.

In this paper we define and characterize cointegrated solutions of continuous-time linear state-space models. A main result is that a cointegrated solution of a continuous-time linear state-space model can be represented as a sum of a Lévy process and a stationary solution of a linear state-space model. Moreover, we prove that the class of cointegrated multivariate Lévy-driven autoregressive moving-average (MCARMA) processes, the continuous-time analogues of the classical vector ARMA processes, is equivalent to the class of cointegrated solutions of continuous-time linear state space models. Necessary conditions for MCARMA processes to be cointegrated are given as well extending the results of Comte [11] for MCAR processes. The conditions depend only on the autoregressive polynomial if we have a minimal model. Finally, we investigate cointegrated continuous-time linear state-space models observed on a discrete time-grid and calculate their linear innovations. Based on the representation of the linear innovations we derive an error correction form. The error correction form uses an infinite linear filter in contrast to the finite linear filter for VAR models.

In this article, we present a robust estimator for the parameters of a stationary, but not necessarily Gaussian, continuous-time ARMA(p, q) (CARMA(p, q)) process that is sampled equidistantly. Therefore, we propose an indirect estimation procedure that first estimates the parameters of the auxiliary AR(r) representation (r ≥ 2p − 1) of the sampled CARMA process using a generalized M-(GM-)estimator. Since the map which maps the parameters of the auxiliary AR(r) representation to the parameters of the CARMA process is not given explicitly, a separate simulation part is necessary where the parameters of the AR(r) representation are estimated from simulated CARMA processes. Then, the parameters which take the minimum distance between the estimated AR parameters and the simulated AR parameters give an estimator for the CARMA parameters. First, we show that under some standard assumptions the GM-estimator for the AR(r) parameters is consistent and asymptotically normally distributed. Then, we prove that the indirect estimator is also consistent and asymptotically normally distributed when the asymptotically normally distributed LS-estimator is used in the simulation part. The indirect estimator satisfies several important robustness properties such as weak resistance, d n -robustness and it has a bounded influence functional. The practical applicability of our method is illustrated in a small simulation study with replacement outliers.

In this paper, we investigate quasi-maximum likelihood (QML) estimation for the parameters of a cointegrated solution of a continuous-time linear state space model observed at discrete time points. The class of cointegrated solutions of continuous-time linear state space models is equivalent to the class of cointegrated continuous-time ARMA (MCARMA) processes. As a start, some pseudo-innovations are constructed to be able to define a QML-function. Moreover, the parameter vector is divided appropriately in long-run and short-run parameters using a representation for cointegrated solutions of continuous-time linear state space models as a sum of a Lévy process plus a stationary solution of a linear state space model. Then, we establish the consistency of our estimator in three steps. First, we show the consistency for the QML estimator of the long-run parameters. In the next step, we calculate its consistency rate. Finally, we use these results to prove the consistency for the QML estimator of the short-run parameters. After all, we derive the limiting distributions of the estimators. The long-run parameters are asymptotically mixed normally distributed, whereas the short-run parameters are asymptotically normally distributed. The performance of the QML estimator is demonstrated by a simulation study.

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