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We study Bayesian networks based on max-linear structural equations as introduced in Gissibl and Klüppelberg [16] and provide a summary of their independence properties. In particular, we emphasize that distributions for such networks are never faithful to the independence model determined by their associated directed acyclic graph unless the latter is a polytree, in which case they are always faithful. In addition, we consider some of the basic issues of estimation and discuss generalized maximum likelihood estimation of the coefficients, using the concept of a generalized likelihood ratio for non-dominated families as introduced by Kiefer and Wolfowitz [21]. Finally, we argue that the structure of a minimal network asymptotically can be identified completely from observational data.

Big data can be a curse and a blessing for the statistician. We report in this paper about some positive effect of big data: big data may open the way for more reliable risk analysis, simply because more extreme data are available. However, big data also require a fully automated analysis. We present here a method, which can easily be implemented and used for large numbers of statistical attributes. We apply this method to safety issues at the landing of airplanes.

We contribute to the understanding of how systemic risk arises in a network of credit-interlinked agents. Motivated by empirical studies we formulate a network model which, despite its simplicity, depicts the nature of interbank markets better than a symmetric model. The components of a vector Ornstein-Uhlenbeck process living on the nodes of the network describe the financial robustnesses of the agents. For this system, we prove a LLN for growing network size leading to a propagation of chaos result. We state properties, which arise from such a structure, and examine the effect of asymmetry on several risk management issues and the possibility of contagion.

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernels to the more general class of regularly varying functions with the corresponding fractional integration parameter. The resulting stochastic processes are called generalized fractional Lévy processes (GFLP) which are shown to exhibit both short and long memory increments possibly with jumps. Moreover, for monotone kernels we define stochastic integrals with respect to GFLPs and investigate their second order structure and path properties. We prove a functional central limit theorem for stochastic integrals driven by a GFLP. As a specific example we present our result for Ornstein-Uhlenbeck processes driven by a time scaled GFLP. This approximation applies to a wide class of stochastic volatility models which include models where neither price nor volatility process are semimartingales.

Understanding and managing risks caused by extreme events is one of the most demanding problems of our society. We consider this topic from a statistical point of view and present some of the probabilistic and statistical theory, which was developed to model and quantify extreme events. By the very nature of an extreme event there will never be enough data to predict a future risk in the classical statistical sense. However, a rather clever probabilistic theory provides us with model classes relevant for the assessment of extreme events. Moreover, specific statistical methods allow for the prediction of rare events, even outside the range of previous observations. We will present the basic theory and relevant examples from climatology (climate change), insurance (return periods of large claims) and finance (portfolio losses and Value-at-Risk estimation).

Financial data are as a rule asymmetric, although most econometric models are symmetric. This applies also to continuous-time models for high-frequency and irregularly spaced data. We discuss some asymmetric Versions of the continuous-time GARCH model, concentrating then on the GJR-COGARCH. We calculate higher order moments and extend the first jump Approximation. These results are prerequisites for Moment estimation and pseudo Maximum likelihood estimation of the GJR-COGRCH Parameters, respectively, which we derive in detail.

Fractional Brownian motion (fBm) can be introduced by a moving average representation driven by standard Brownian motion, which is an affine Markov process. Motivated by this we aim at results analogous to those achieved in recent years for affine models. Using a simple prediction formula for the conditional expectation of a fBm and its Gaussianity, we calculate the conditional characteristic functions of fBm and related processes, including important examples like fractional Ornstein-Uhlenbeck- or Cox-Ingersoll-Ross processes. As an application we propose a fractional Vasicek bond market model and compare prices of zero coupon bonds to those achieved in the classical Vasicek case.

Based on the concept of a Lévy copula to describe the dependence structure of a multi-variate Lévy process, we present a new estimation procedure. We consider a parametric model for the marginal Lévy processes as well as for the Lévy copula and estimate the parameters by a two-step procedure. We first estimate the parameters of the marginal processes and then estimate in a second step only the dependence structure parameter. For infinite Lévy measures, we truncate the small jumps and base our statistical analysis on the large jumps of the model. Prominent example will be a bivariate stable Lévy process, which allows for analytic calculations and, hence, for a comparison of different methods. We prove asymptotic normality of the parameter estimates from the two-step procedure, and in particular, we derive the Godambe information matrix, whose inverse is the covariance matrix of the normal limit law. A simulation study investigates the loss of efficiency because of the two-step procedure and the truncation.

Motivated by recent developments in light of the sub-prime and subsequent financial crisis we fit two different vector autoregressive generalized conditional heteroscedastic (VAR-GARCH) models to three financial indices with the aim of understanding the development of dependency structures between credit spreads and other macroeconomic variables. Our analysis includes daily quotes from June 2004 to April 2009 of the iTraxx Europe index, the Dow Jones Euro Stoxx 50 index, and the Dow Jones VStoxx index. We propose a robust, time-varying modeling approach concerning the conditional mean, and a BEKK versus DCC-GARCH approach concerning the conditional covariance. Furthermore we allow for a parsimonious model specification by setting insignificant coefficients to zero. Our empirical results indicate that the autoregressive coefficients vary strongly with time and even change their signs. Well-known interrelations, such as the negative correlation between CDS’ and stocks are lost through the financial crisis. The conditional covariance estimates in the BEKK and DCC model are fairly similar, given the difference in the number of model parameters. We found evidence of strongly varying conditional variances and correlations, with dependencies increasing after the outbreak of the financial crisis. This knowledge may help to improve decision tools in the financial industry, especially in areas such as asset pricing, portfolio selection, and risk management.

Interest in continuous-time processes has increased rapidly in recent years, largely because of the high-frequency data available in many areas of application, particularly in finance and turbulence. We develop a method for estimating the kernel function of a continuous-time moving average (CMA) process Y which takes advantage of the high-frequency of the data. In order to do so we examine the relation between the CMA process Y and the discrete-time process $Y^\Delta$ obtained by sampling Y at times which are integer multiples of some small positive $\Delta$. In particular we derive asymptotic results as $\Delta\downarrow 0$ which generalize results of Brockwell, Ferrazzano and Klüppelberg (2011) for high-frequency sampling of CARMA processes. We propose an estimator of the continuous-time kernel based on observations of $Y^\Delta$, investigate its properties and illustrate its performance using simulated data. Particular attention is paid to the performance of the estimator as $\Delta\downarrow 0$. Time-domain and frequency-domain methods are used to obtain insight into CMA processes and their sampled versions.

Large insurance losses happen infrequently, but they do happen. In this paper we present the standard distribution models used in fire, wind–storm or flood insurance and mention some insurance applications.

Dependence modelling and estimation is a key issue in the assessment of financial risk. It is common knowledge meanwhile that the multivariate normal model with linear correlation as its natural dependence measure is by no means an ideal model. We suggest a large class of models and a dependence function, which allows us to capture the complete extreme dependence structure of a portfolio. We also present a simple nonparametric estimation procedure of this function. To show our new method at work we apply it to a financial data set of high frequency stock data and estimate the extreme dependence in the data. Among the results in the investigation we show that the extreme dependence is the same for different time scales. This is consistent with the result on high frequency FX data reported in Hauksson et al. (2001).Hence, the two different asset classes seem to share the same time scaling for extreme dependency. This time scaling property of high frequency data is also explained from a theoretical point of view.

Sums of Lévy-driven Ornstein-Uhlenbeck processes seem appropriate for modelling electricity spot price data. In this paper we present a new estimation method with particular emphasis on capturing the high peaks, which is one of the stylized features of such data. After introducing our method we show it at work for the EEX Phelix Base electricity price index. We also present a small simulation study to show the performance of our estimation procedure.

In Böcker and Klüppelberg (2005) we presented a simple approximation of Op- VaR of a single operational risk cell. The present paper derives approximations of similar quality and simplicity for the multivariate problem. Our approach is based on modelling of the dependence structure of different cells via the new concept of a Lévy copula.

We present several notions of high-level dependence for stochastic processes, which have appeared in the literature. We calculate such measures for discrete and continuous-time models, where we concentrate on time series with heavy-tailed marginals, where extremes are likely to occur in clusters. Such models include linear models and solutions to random recurrence equations; in particular, discrete and continuous-time moving average and (G)ARCH processes. To illustrate our results we present a small simulation study.

Private equity firms are blamed for quickly extracting all of a target company's cash, and sometimes for going even further by requesting a target company to incur additional debts { in order to be able to pay investors an additional dividend { and thus driving it into bankruptcy. Consequently, there is always a trade-off between the benefit of high extra dividends and the associated risk, due to higher debt obligations, which may cause bankruptcy. In this paper, we apply real-options theory and capital-budgeting techniques to the problem of assessing a private investor's risk. We propose a new continuous time DCF model, which incorporates the four fundamental value drivers, among others a high debt to equity ratio, and typical characteristics for private investments like a high probability to default. We also introduce different risk measures by proposing a new measurement model that is based on the cash flow process, the associated multiple process as well as on the implied IRR of the transaction. Finally, we give some details of how such a model is implemented to provide investors as well as debt lenders with a decision support regarding different investments or investments strategies.

We investigate optimal consumption problems for a Black–Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall for logarithmic utility functions. We find the solutions in terms of a dynamic strategy in explicit form, which can be compared and interpreted. This paper continues our previous work, where we solved similar problems for power utility functions.

We investigate optimal consumption and investment problems for a Black- Scholes market under uniform restrictions on Value-at-Risk and Expected Shortfall. We formulate various utility maximization problems,which can be solved explicitly. We compare the optimal solutions in form of optimal value, optimal control and optimal wealth to analogous problems under additional uniform risk bounds. Our proofs are partly based on solutions to Hamilton-Jacobi-Bellman equations, and we prove a corresponding verification theorem.

Wir untersuchen das Problem der Modellierung und Quantifizierung des multivariaten operationellen Risikos. Basierend auf dem weit verbreiteten eindimensionalen Verlustverteilungs-Ansatz, entwickeln wir zunächst ein "Invarianzprinzip", dem jedes multivariate Modell genügen sollte. Unser Ansatz basiert auf dem neuen Konzept der Pareto- Lévy-Copula und erlaubt für das operationelle Risiko eine dynamische Betrachtungsweise zu jedem beliebigen zukünftigen Zeitpunkt. Dabei nutzen wir die Tatsache, dass operationelle Verlustdaten üblicherweise langschwänzig ("heavy tailed") sind und deshalb hervorragend durch das Konzept der multivariaten regulären Variation beschrieben werden können. Schließlich leiten wir für wichtige Beispiele der Pareto-Lévy-Copula und der Verlusthöhenverteilung analytische Approximationen für den operationellen Value-at-Risk her.

The discrete time ARCH/GARCH model of Engle and Bollarslev has been enormously influential and successful in the modelling of financial data. Recently, Kl¨uppelberg, Lindner, and Maller (2004) introduced the so-called “COGARCH” model as a continuoustime analogue to the GARCH model. Many aspects of the COGARCH have been investigated, including various of its theoretical properties, its relations to other continuous-time models, and the estimation of the parameters in it. We review some of these results in the present paper, and go on to apply the COGARCH to 5-minute data on the S&P500 index, in order to illustrate its ability to analyse stochastic volatility in very high-frequency, irregularly spaced, financial data.

We consider an insurance risk model for the cashflow of an insurance company, which invests its reser.v.e into a portfolio consisting of risky and riskless assets. The price of the risky asset is modeled by an exponential Lévy process. We derive the integrated risk process and the corresponding discounted net loss process. We calculate certain quantities as characteristic functions and moments. We also show under weak conditions stationarity of the discounted net loss process and derive the left and right tail behaviour of the model. Our results show that the model carries a high risk, which may originate either from large insurance claims or from the risky investment.

Simultaneous modelling of operational risks occurring in different event type/business line cells poses a serious challenge for operational risk quantification. Here we invoke the new concept of Lévy copulas to model the dependence structure of operational loss events. We explain the consequences of this dependence concept for frequencies and severities of operational risk in detail. For important examples of the Lévy copula and heavy-tailed GPD tail severities we derive first order approximations for multivariate operational VAR.

In general, the risk of joint extreme outcomes in financial markets can be expressed as a function of the tail dependence function of a high-dimensional vector after standardizing marginals. Hence it is of importance to model and estimate tail dependence functions. Even for moderate dimension, nonparametrically estimating a tail dependence function is very inefficient and fitting a parametric model to tail dependence functions is not robust. In this paper we propose a semi-parametric model for (asymptotically dependent) tail dependence functions via an elliptical copula. Based on this model assumption, we propose a novel estimator for the tail dependence function, which proves favorable compared to the empirical tail dependence function estimator, both theoretically and empirically.

In this paper we investigate an optimal investment and consumption problem for an investor who trades in a Black-Scholes financial market with stochastic coefficients driven by a non-Gaussian Ornstein-Uhlenbeck process. We assume that an agent makes consumption and investment decisions based on a HARA utility function. By applying the usual separation method in the variables, we are faced with the problem of solving a non-linear (semilinear) first-order partial integro-differential equation. A candidate solution is derived via the Feynman-Kac representation. By using the properties of an operator, defined in a suitable functional space, we prove uniqueness and smoothness of the solution. Optimality is verified by applying a classical verification theorem.

We consider an insurance risk process with the possibility to invest the capital reserve into a portfolio consisting of risky assets and a riskless asset. The stock price is modeled by an exponential Lévy process and the riskless interest rate is assumed to be constant. We aim at the risk assessment of the integrated risk process in terms of a high quantile or the far out distribution tail. We indicate an application to an optimal investment strategy of an insurer.

We estimate conditional and unconditional high quantiles for electricity spot prices based on a linear model with stable innovations. This approach captures the impressive peaks in such data and, as a four-parametric family captures also the asymmetry in the innovations. Moreover, it allows for explicit formulas of quantiles, which can then be calculated recursively from day to day. We also prove that conditional quantiles of step h ε N converge for h→∞ to the corresponding unconditional quantiles. The paper is motivated by the daily spot prices from the Singapore New Electricity Market, which serves an example to show our method at work.

Recently there has been growing interest in applying elliptical distributions to risk management. Under certain conditions, Hult and Lindskog show that a random vector with an elliptical distribution is in the domain of attraction of a multivariate extreme value distribution. In this paper we study two estimators for the tail dependence function, which are based on extreme value theory and the structure of an elliptical distribution. After deriving second-order regular variation estimates and proving asymptotic normality for both estimators, we show that the estimator based on the structure of an elliptical distribution is better than that based on extreme value theory in terms of both asymptotic variance and optimal asymptotic mean squared error. Our simulation study further confirms this.

Barndorff-Nielsen and Shephard [3] investigate supOU processes as volatility models. Empirical volatility has tails heavier than normal, long memory in the sense that the empirical autocorrelation function decreases slower than exponential, and exhibits volatility clusters on high levels. We investigate supOU processes with respect to these stylized facts. The class of supOU processes is vast and can be distinguished by its underlying driving Lévy process. Within the classes of convolution equivalent distributions we shall show that extremal clusters and long range dependence only occur for supOU processes, whose underlying driving Lévy process has regularly varying increments. The results on the extremal behavior of supOU processes correspond to the results of classical Lévy-driven OU processes.