Hans-Peter Scheffler

A second-order accurate numerical approximation for the fractional diffusion equation

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. In this paper, we examine a practical numerical method which is second-order accurate in time and in space to solve a class of initial-boundary value fractional diffusive equations with variable coefficients on a finite domain. An approach based on the classical Crank-Nicholson method combined with spatial extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Stability, consistency, and (therefore) convergence of the method are examined. It is shown that the fractional Crank-Nicholson method based on the shifted Grunwald formula is unconditionally stable. A numerical example is presented and compared with the exact analytical solution for its order of convergence.

A simple robust estimation method for the thickness of heavy tails

We present a simple general method for estimating the thickness of heavy tails based on the asymptotics of the sum. The method works for dependent data, and only requires that the centered and normalized partial sums are stochastically compact. For data in the domain of attraction of a stable law our estimator is asymptotically log stable, consistent and asymptotically unbiased, and converges in the mean-square sense to the index of regular variation.

Portfolio Modeling with Heavy Tailed Random Vectors

Since the work of Mandelbrot in the 1960s there has accumulated a great deal of empirical evidence for heavy tailed models in finance. In these models, the probability of a large fluctuation falls off like a power law. The generalized central limit theorem shows that these heavy-tailed fluctuations accumulate to a stable probability distribution. If the tails are not too heavy then the variance is finite and we find the familiar normal limit, a special case of stable distributions. Otherwise the limit is a nonnormal stable distribution, whose bell-shaped density may be skewed, and whose probability tails fall off like a power law. The most important model parameter for such distributions is the tail thickness α, which governs the rate at which the probability of large fluctuations diminishes. A smaller value of α means that the probability tails are fatter, implying more volatility. In fact, when α < 2 the theoretical variance is infinite. A portfolio can be modeled using random vectors, where each entry of the vector represents a different asset. The tail parameter α usually depends on the coordinate. The wrong coordinate system can mask variations in α, since the heaviest tail tends to dominate. A judicious choice of coordinate system is given by the eigenvectors of the sample covariance matrix. This isolates the heaviest tails, associated with the largest eigenvalues, and allows a more faithful representation of the dependence between assets.

Fractional governing equations for coupled random walks

In a continuous time random walk (CTRW), a random waiting time precedes each random jump. The CTRW is coupled if the waiting time and the subsequent jump are dependent random variables. The CTRW is used in physics to model diffusing particles. Its scaling limit is governed by an anomalous diffusion equation. Some applications require an overshoot continuous time random walk (OCTRW), where the waiting time is coupled to the previous jump. This paper develops stochastic limit theory and governing equations for CTRW and OCTRW. The governing equations involve coupled space-time fractional derivatives. In the case of infinite mean waiting times, the solutions to the CTRW and OCTRW governing equations can be quite different.

The domains of partial attraction of probabilities on groups and on vector spaces

It is shown that for exponential Lie groupsG the limit behavior of i.i.d. triangular arrays on the groupG and on the tangent spaceG coincide. This result is used to obtain a characterization of domains of partial attraction (resp. semistable attraction) on exponential (resp. simply connected nilpotent) Lie groups via the corresponding domains on the tangent space.

Moving Averages of Random Vectors with Regularly Varying Tails

Regular variation is an analytic condition on the tails of a probability distribution which is necessary for an extended central limit theorem to hold, when the tails are too heavy to allow attraction to a normal limit. The limiting distributions which can occur are called operator stable. In this paper we show that moving averages of random vectors with regularly varying tails are in the generalized domain of attraction of an operator stable law. We also prove that the sample autocovariance matrix of these moving averages is in the generalized domain of attraction of an operator stable law on the vector space of symmetric matrices. AMS 1990 subject classification. Primary 62M10, secondary 62E20, 62F12, 60F05.

Sample Cross‐correlations for Moving Averages with Regularly Varying Tails

We compute the asymptotics of the sample cross‐correlation between two scalar moving averages whose IID innovations have regularly varying probability tails with different tail indices.

A law of the iterated logarithm for heavy-tailed random vectors

Abstract. For a random vector belonging to the (generalized) domain of operator semistable attraction of some nonnormal law we prove various variants of Chovers law of the iterated logarithm for the partial sum. Furthermore we also derive some large deviation results necessary for the proof of our main theorems.

Multivariable Regular Variation of Functions and Measures

Abstract Regular variation is an asymptotic property of functions and measures. The one variable theory is well–established, and has found numerous applications in both pure and applied mathematics. In this paper we present several new results on multivariable regular variation for functions and measures.

Spectral Decomposition for Generalized Domains of Semistable Attraction

AbstractSuppose X, X1, X2, X3,... are i.i.d. random vectors, and kn a sequence of positive integers tending to infinity in such a way that kn+1/kn→c≥1. If there exist linear operators An and constant vectors bn such that