arXiv subjects

Claudia Klüppelberg

Publications and source records attributed to Claudia Klüppelberg.

At least 37 records · Page 2Linked to original sources

arXiv2017

Fractionally integrated COGARCH processes

We construct fractionally integrated continuous-time GARCH models, which capture the observed long range dependence of squared volatility in high-frequency data. Since the usual Molchan-Golosov and Mandelbrot-van-Ness fractional kernels lead to problems in the definition of the model, we resort to moderately long memory processes by choosing a fractional parameter $d\in(-0.5,0)$ and remove the singularities of the kernel to obtain non-pathological sample paths. The volatility of the new fractional COGARCH process has positive features like stationarity, and its covariance function shows an algebraic decay, which make it applicable to econometric high-frequency data. In an empirical application the model is fitted to exchange rate data using a simulation-based version of the generalised method of moments.

math.ST↗

arXiv2017

Tail dependence of recursive max-linear models with regularly varying noise variables

Recursive max-linear structural equation models with regularly varying noise variables are considered. Their causal structure is represented by a directed acyclic graph (DAG). The problem of identifying a recursive max-linear model and its associated DAG from its matrix of pairwise tail dependence coefficients is discussed. For example, it is shown that if a causal ordering of the associated DAG is additionally known, then the minimum DAG representing the recursive structural equations can be recovered from the tail dependence matrix. For the relevant subclass of recursive max-linear models, identifiability of the associated minimum DAG from the tail dependence matrix and the initial nodes is shown. Algorithms find the associated minimum DAG for the different situations. Furthermore, given a tail dependence matrix, an algorithm outputs all compatible recursive max-linear models and their associated minimum DAGs.

stat.ME↗

arXiv2017

Contagion in financial systems: A Bayesian network approach

We develop a structural default model for interconnected financial institutions in a probabilistic framework. For all possible network structures we characterize the joint default distribution of the system using Bayesian network methodologies. Particular emphasis is given to the treatment and consequences of cyclic financial linkages. We further demonstrate how Bayesian network theory can be applied to detect contagion channels within the financial network, to measure the systemic importance of selected entities on others, and to compute conditional or unconditional probabilities of default for single or multiple institutions.

q-fin.RM↗

arXiv2017

Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional

We prove rigorously that the exact N-electron Hohenberg-Kohn density functional converges in the strongly interacting limit to the strictly correlated electrons (SCE) functional, and that the absolute value squared of the associated constrained-search wavefunction tends weakly in the sense of probability measures to a minimizer of the multi-marginal optimal transport problem with Coulomb cost associated to the SCE functional. This extends our previous work for N=2 [CFK11]. The correct limit problem has been derived in the physics literature by Seidl [Se99] and Seidl, Gori-Giorgi and Savin [SGS07]; in these papers the lack of a rigorous proof was pointed out. We also give a mathematical counterexample to this type of result, by replacing the constraint of given one-body density -- an infinite-dimensional quadratic expression in the wavefunction -- by an infinite-dimensional quadratic expression in the wavefunction and its gradient. Connections with the Lawrentiev phenomenon in the calculus of variations are indicated.

math-ph↗

arXiv2017

Limit theory for the empirical extremogram of random fields

Regularly varying stochastic processes are able to model extremal dependence between process values at locations in random fields. We investigate the empirical extremogram as an estimator of dependence in the extremes. We provide conditions to ensure asymptotic normality of the empirical extremogram centred by a pre-asymptotic version. For max-stable processes with Fr{é}chet margins we provide conditions such that the empirical extremogram centred by its true version is asymptotically normal. The results of this paper apply to a variety of spatial and space-time processes, and to time series models. We apply our results to max-moving average processes and Brown-Resnick processes.

math.ST↗

arXiv2017

Partial mean field limits in heterogeneous networks

We investigate systems of interacting stochastic differential equations with two kinds of heterogeneity: one originating from different weights of the linkages, and one concerning their asymptotic relevance when the system becomes large. To capture these effects we define a partial mean field system, and prove a law of large numbers with explicit bounds on the mean squared error. Furthermore, a large deviation result is established under reasonable assumptions. The theory will be illustrated by several examples: on the one hand, we recover the classical results of chaos propagation for homogeneous systems, and on the other hand, we demonstrate the validity of our assumptions for quite general heterogeneous networks including those arising from preferential attachment random graph models.

math.PR↗

arXiv2017

Max-linear models on directed acyclic graphs

We consider a new recursive structural equation model where all variables can be written as max-linear function of their parental node variables and independent noise variables. The model is max-linear in terms of the noise variables, and its causal structure is represented by a directed acyclic graph. We detail the relation between the weights of the recursive structural equation model and the coefficients in its max-linear representation. In particular, we characterize all max-linear models which are generated by a recursive structural equation model, and show that its max-linear coefficient matrix is the solution of a fixed point equation. We also find a unique minimum directed acyclic graph representing the recursive structural equations of the variables. The model structure introduces a natural order between the node variables and the max-linear coefficients. This yields representations of the vector components, which are based on a minimum number of node and noise variables.

math.PR↗

arXiv2016

Conditional loss probabilities for systems of economic agents sharing light-tailed claims with analysis of portfolio diversification benefits

We analyze systems of agents sharing light-tailed risky claims issued by different financial objects. Assuming exponentially distributed claims, we obtain that both agents' and system's losses follow generalized exponential mixture distributions. We show that this leads to qualitatively different results on individual and system risks compared to heavy-tailed claims previously studied in the literature. By deducing conditional loss distributions we investigate the impact of stress situations on agents' and system's losses. Moreover, we present a criterion for agents to decide whether holding few objects or portfolio diversification minimizes their risks in system crisis situations.

q-fin.RM↗

arXiv2016

An Innovations Algorithm for the prediction of functional linear processes

When observations are curves over some natural time interval, the field of functional data analysis comes into play. Functional linear processes account for temporal dependence in the data. The prediction problem for functional linear processes has been solved theoretically, but the focus for applications has been on functional autoregressive processes. We propose a new computationally tractable linear predictor for functional linear processes. It is based on an application of the Multivariate Innovations Algorithm to finite-dimensional subprocesses of increasing dimension of the infinite-dimensional functional linear process. We investigate the behavior of the predictor for increasing sample size. We show that, depending on the decay rate of the eigenvalues of the covariance and the spectral density operator, the resulting predictor converges with a certain rate to the theoretically best linear predictor.

stat.ME↗

arXiv2016

Anisotropic Brown-Resnick space-time processes: estimation and model assessment

Spatially isotropic max-stable processes have been used to model extreme spatial or space-time observations. One prominent model is the Brown-Resnick process, which has been successfully fitted to time series, spatial data and space-time data. This paper extends the process to possibly anisotropic spatial structures. For regular grid observations we prove strong consistency and asymptotic normality of pairwise maximum likelihood estimates for fixed and increasing spatial domain, when the number of observations in time tends to infinity. We also present a statistical test for isotropy versus anisotropy. We apply our test to precipitation data in Florida, and present some diagnostic tools for model assessment. Finally, we present a method to predict conditional probability fields and apply it to the data.

stat.ME↗

arXiv2016

Passage time and fluctuation calculations for subexponential Lévy processes

We consider the passage time problem for Lévy processes, emphasising heavy tailed cases. Results are obtained under quite mild assumptions, namely, drift to $-\infty$ a.s. of the process, possibly at a linear rate (the finite mean case), but possibly much faster (the infinite mean case), together with subexponential growth on the positive side. Local and functional versions of limit distributions are derived for the passage time itself, as well as for the position of the process just prior to passage, and the overshoot of a high level. A significant connection is made with extreme value theory via regular variation or maximum domain of attraction conditions imposed on the positive tail of the canonical measure, which are shown to be necessary for the kind of convergence behaviour we are interested in.

math.PR↗

arXiv2016

Time-consistency of risk measures with GARCH volatilities and their estimation

In this paper we study time-consistent risk measures for returns that are given by a GARCH(1,1) model. We present a construction of risk measures based on their static counterparts that overcomes the lack of time-consistency. We then study in detail our construction for the risk measures Value-at-Risk (VaR) and Average Value-at-Risk (AVaR). While in the VaR case we can derive an analytical formula for its time-consistent counterpart, in the AVaR case we derive lower and upper bounds to its time-consistent version. Furthermore, we incorporate techniques from Extreme Value Theory (EVT) to allow for a more tail-geared statistical analysis of the corresponding risk measures. We conclude with an application of our results to a data set of stock prices.

q-fin.RM↗

arXiv2015

Conditional risk measures in a bipartite market structure

In this paper we study the effect of network structure between agents and objects on measures for systemic risk. We model the influence of sharing large exogeneous losses to the financial or (re)insuance market by a bipartite graph. Using Pareto-tailed losses and multivariate regular variation we obtain asymptotic results for systemic conditional risk measures based on the Value-at-Risk and the Conditional Tail Expectation. These results allow us to assess the influence of an individual institution on the systemic or market risk and vice versa through a collection of conditional systemic risk measures. For large markets Poisson approximations of the relevant constants are provided in the example of an insurance market. The example of an underlying homogeneous random graph is analysed in detail, and the results are illustrated through simulations.

q-fin.RM↗

arXiv2015

Integrability conditions for space-time stochastic integrals: Theory and applications

We derive explicit integrability conditions for stochastic integrals taken over time and space driven by a random measure. Our main tool is a canonical decomposition of a random measure which extends the results from the purely temporal case. We show that the characteristics of this decomposition can be chosen as predictable strict random measures, and we compute the characteristics of the stochastic integral process. We apply our conditions to a variety of examples, in particular to ambit processes, which represent a rich model class.

math.PR↗

arXiv2015

Simulation of stochastic Volterra equations driven by space--time Lévy noise

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space--time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods $L^p$- and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

math.PR↗

arXiv2014

Superposition of COGARCH processes

We suggest three superpositions of COGARCH (supCOGARCH) volatility processes driven by Lévy processes or Lévy bases. We investigate second-order properties, jump behaviour, and prove that they exhibit Pareto-like tails. Corresponding price processes are defined and studied. We find that the supCOGARCH models allow for more flexible autocovariance structures than the COGARCH. Moreover, other than most financial volatility models, the supCOGARCH processes do not exhibit a deterministic relationship between price and volatility jumps. Furthermore, in one supCOGARCH model not all volatility jumps entail a price jump, while in another supCOGARCH model not all price jumps necessarily lead to volatility jumps.

math.PR↗

arXiv2014

Systemic risk through contagion in a core-periphery structured banking network

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 homogeneous model. The components of a vector Ornstein-Uhlenbeck process living on the vertices 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 inhomogeneity on several risk management issues and the possibility of contagion.

q-fin.RM↗

arXiv2014

Asymmetric COGARCH processes

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-COGARCH parameters, respectively, which we derive in detail.

math.ST↗