arXiv subjects

Claudia Klüppelberg

Publications and source records attributed to Claudia Klüppelberg.

49 records · Page 3Linked to original sources

arXiv2013

Two-step estimation of a multivariate Lévy process

Based on the concept of a Lévy copula to describe the dependence structure of a multivariate 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 \lp, 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.

stat.ME↗

arXiv2013

N-density representability and the optimal transport limit of the Hohenberg-Kohn functional

We derive and analyze a hierarchy of approximations to the strongly correlated limit of the Hohenberg-Kohn functional. These "density representability approximations" are obtained by first noting that in the strongly correlated limit, N-representability of the pair density reduces to the requirement that the pair density must come from a symmetric N-point density. One then relaxes this requirement to the existence of a representing symmetric k-point density with k < N. The approximate energy can be computed by simulating a fictitious k-electron system. We investigate the approximations by deriving analytically exact results for a 2-site model problem, and by incorporating them into a self-consistent Kohn-Sham calculation for small atoms. We find that the low order representability conditions already capture the main part of the correlations.

physics.chem-ph↗

arXiv2013

High-frequency sampling and kernel estimation for continuous-time moving average processes

Interest in continuous-time processes has increased rapidly in recent years, largely because of high-frequency data available in many applications. We develop a method for estimating the kernel function $g$ of a second-order stationary Lévy-driven continuous-time moving average (CMA) process $Y$ based on observations of the discrete-time process $Y^Δ$ obtained by sampling $Y$ at $Δ, 2Δ,...,nΔ$ for small $Δ$. We approximate $g$ by $g^Δ$ based on the Wold representation and prove its pointwise convergence to $g$ as $Δ\rightarrow 0$ for $\CARMA(p,q)$ processes. Two non-parametric estimators of $g^Δ$, based on the innovations algorithm and the Durbin-Levinson algorithm, are proposed to estimate $g$. For a Gaussian CARMA process we give conditions on the sample size $n$ and the grid-spacing $Δ(n)$ under which the innovations estimator is consistent and asymptotically normal as $n\rightarrow\infty$. The estimators can be calculated from sampled observations of {\it any} CMA process and simulations suggest that they perform well even outside the class of CARMA processes. We illustrate their performance for simulated data and apply them to the Brookhaven turbulent wind speed data. Finally we extend results of \citet{bfk:2011:1} for sampled CARMA processes to a much wider class of CMA processes.

math.ST↗

arXiv2012

Turbulence modeling by time-series methods

A general model for stationary, time-wise turbulent velocity is presented and discussed. This approach, inspired by modeling ideas of Barndorff-Nielsen and Schimgel, is coherent with the K41 hypothesis of local isotropy, and it allows us to separate second-order statistics from higher order ones. The model can be motivated by Taylor's hypothesis and a relation between time and spatial spectra. Second order statistics are used to separate the deterministic kernel function and the weakly stationary driving noise. A non-parametric estimation method for the turbulence intermittency is suggested.

physics.data-an↗

arXiv2012

Statistical inference for max-stable processes in space and time

Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several representations of max-stable random fields have been proposed in the literature. One such representation is based on a limit of normalized and scaled pointwise maxima of stationary Gaussian processes that was first introduced by Kabluchko, Schlather and de Haan (2009). This paper deals with statistical inference for max-stable space-time processes that are defined in an analogous fashion. We describe pairwise likelihood estimation, where the pairwise density of the process is used to estimate the model parameters and prove strong consistency and asymptotic normality of the parameter estimates for an increasing space-time dimension, i.e., as the joint number of spatial locations and time points tends to infinity. A simulation study shows that the proposed method works well for these models.

stat.ME↗

arXiv2012

Futures pricing in electricity markets based on stable CARMA spot models

We present a new model for the electricity spot price dynamics, which is able to capture seasonality, low-frequency dynamics and the extreme spikes in the market. Instead of the usual purely deterministic trend we introduce a non-stationary independent increments process for the low-frequency dynamics, and model the large fluctuations by a non-Gaussian stable CARMA process. The model allows for analytic futures prices, and we apply these to model and estimate the whole market consistently. Besides standard parameter estimation, an estimation procedure is suggested, where we fit the non-stationary trend using futures data with long time until delivery, and a robust $L^1$-filter to find the states of the CARMA process. The procedure also involves the empirical and theoretical risk premiums which -- as a by-product -- are also estimated. We apply this procedure to data from the German electricity exchange EEX, where we split the empirical analysis into base load and peak load prices. We find an overall negative risk premium for the base load futures contracts, except for contracts close to delivery, where a small positive risk premium is detected. The peak load contracts, on the other hand, show a clear positive risk premium, when they are close to delivery, while the contracts in the longer end also have a negative premium.

stat.AP↗

arXiv2011

Max-stable processes for modelling extremes observed in space and time

Max-stable processes have proved to be useful for the statistical modelling of spatial extremes. Several representations of max-stable random fields have been proposed in the literature. For statistical inference it is often assumed that there is no temporal dependence, i.e., the observations at spatial locations are independent in time. We use two representations of stationary max-stable spatial random fields and extend the concepts to the space-time domain. In a first approach, we extend the idea of constructing max-stable random fields as limits of normalized and rescaled pointwise maxima of independent Gaussian random fields, which was introduced by Kabluchko, Schlather and de Haan [2009], who construct max-stable random fields associated to a class of variograms. We use a similar approach based on a well-known result by Hüsler and Reiss and apply specific spatio-temporal covariance models for the underlying Gaussian random field, which satisfy weak regularity assumptions. Furthermore, we extend Smith's storm profile model to a space-time setting and provide explicit expressions for the bivariate distribution functions. The tail dependence coefficient is an important measure of extremal dependence. We show how the spatio-temporal covariance function underlying the Gaussian random field can be interpreted in terms of the tail dependence coefficient. Within this context, we examine different concepts for constructing spatio-temporal covariance models and analyse several specific examples, including Gneiting's class of nonseparable stationary covariance functions.

stat.ME↗

arXiv2011

Density functional theory and optimal transportation with Coulomb cost

We present here novel insight into exchange-correlation functionals in density functional theory, based on the viewpoint of optimal transport. We show that in the case of two electrons and in the semiclassical limit, the exact exchange-correlation functional reduces to a very interesting functional of novel form, which depends on an optimal transport map $T$ associated with a given density $ρ$. Since the above limit is strongly correlated, the limit functional yields insight into electron correlations. We prove the existence and uniqueness of such an optimal map for any number of electrons and each $ρ$, and determine the map explicitly in the case when $ρ$ is radially symmetric.

math.AP↗

arXiv2011

High frequency sampling of a continuous-time ARMA process

Continuous-time autoregressive moving average (CARMA) processes have recently been used widely in the modeling of non-uniformly spaced data and as a tool for dealing with high-frequency data of the form $Y_{nΔ}, n=0,1,2,...$, where $Δ$ is small and positive. Such data occur in many fields of application, particularly in finance and the study of turbulence. This paper is concerned with the characteristics of the process $(Y_{nΔ})_{n\in\bbz}$, when $Δ$ is small and the underlying continuous-time process $(Y_t)_{t\in\bbr}$ is a specified CARMA process.

math.ST↗

arXiv2011

Fractional Lévy-driven Ornstein--Uhlenbeck processes and stochastic differential equations

Using Riemann-Stieltjes methods for integrators of bounded $p$-variation we define a pathwise integral driven by a fractional Lévy process (FLP). To explicitly solve general fractional stochastic differential equations (SDEs) we introduce an Ornstein-Uhlenbeck model by a stochastic integral representation, where the driving stochastic process is an FLP. To achieve the convergence of improper integrals, the long-time behavior of FLPs is derived. This is sufficient to define the fractional Lévy-Ornstein-Uhlenbeck process (FLOUP) pathwise as an improper Riemann-Stieltjes integral. We show further that the FLOUP is the unique stationary solution of the corresponding Langevin equation. Furthermore, we calculate the autocovariance function and prove that its increments exhibit long-range dependence. Exploiting the Langevin equation, we consider SDEs driven by FLPs of bounded $p$-variation for $p<2$ and construct solutions using the corresponding FLOUP. Finally, we consider examples of such SDEs, including various state space transforms of the FLOUP and also fractional Lévy-driven Cox-Ingersoll-Ross (CIR) models.

math.ST↗

arXiv2011

On the Ruin Probability of the Generalised Ornstein-Uhlenbeck Process in the Cramér Case

For a bivariate \Levy process $(ξ_t,η_t)_{t\ge 0}$ and initial value $V_0$ define the Generalised Ornstein-Uhlenbeck (GOU) process \[ V_t:=e^{ξ_t}\Big(V_0+\int_0^t e^{-ξ_{s-}}\ud η_s\Big),\quad t\ge0,\] and the associated stochastic integral process \[Z_t:=\int_0^t e^{-ξ_{s-}}\ud η_s,\quad t\ge0.\] Let $T_z:=\inf\{t>0:V_t<0\mid V_0=z\}$ and $ψ(z):=P(T_z<\infty)$ for $z\ge 0$ be the ruin time and infinite horizon ruin probability of the GOU. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for $ψ(z)$ and the distribution of $T_z$ as $z\to\infty$, under very general, easily checkable, assumptions, when $ξ$ satisfies a Cramér condition.

math.PR↗

arXiv2009

The first passage event for sums of dependent Lévy processes with applications to insurance risk

For the sum process $X=X^1+X^2$ of a bivariate Lévy process $(X^1,X^2)$ with possibly dependent components, we derive a quintuple law describing the first upwards passage event of $X$ over a fixed barrier, caused by a jump, by the joint distribution of five quantities: the time relative to the time of the previous maximum, the time of the previous maximum, the overshoot, the undershoot and the undershoot of the previous maximum. The dependence between the jumps of $X^1$ and $X^2$ is modeled by a Lévy copula. We calculate these quantities for some examples, where we pay particular attention to the influence of the dependence structure. We apply our findings to the ruin event of an insurance risk process.

math.PR↗

arXiv2008

Optimal investment and consumption in a Black--Scholes market with Lévy-driven stochastic coefficients

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 investment and consumption decisions based on a power utility function. By applying the usual separation method in the variables, we are faced with the problem of solving a nonlinear (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 function space, we prove uniqueness and smoothness of the solution. Optimality is verified by applying a classical verification theorem.

q-fin.PR↗