Table of contents, Bernoulli Journal, vol. 1, no. 3 (1995)

Stable fractal sums of pulses: the cylindrical case; Authors: Renata Cioczek-Georges, Benoit B. Mandelbrot, Gennady Samorodnitsky, Murad S. Taqqu

Supports of doubly stochastic measures; Authors: Kevin Hestir, Stanley C. Williams

Joint distribution of maxima of concomitants of subsets of order statistics; Authors: S.N. Joshi, H.N. Nagaraja

Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes; Authors: Asger Roer Pedersen

Hyperbolic distributions in finance; Authors: Ernst Eberlein, Ulrich Keller

Editorial board, Bernoulli Journal, vol. 1, no. 3 (1995)

Ole E. Barndorff-Nielsen, Editor-in-Chief

Jens Ledet Jensen, Managing Co-Editor

Peter J. Bickel, Editor

David Nualart, Editor

David R. Cox, Editor

Edward C. Waymire, Editor

Lucien Birgé, Associate Editor

Louis H. Y. Chen, Associate Editor

Hans Föllmer, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Dag Tjøstheim, Associate Editor

A class of α-stable, 0<α<2, processes is obtained as a sum of 'up-and-down' pulses determined by an appropriate Poisson random measure. Processes are H-self-affine (also frequently called 'self-similar') with H<1/α and have stationary increments. Their two-dimensional dependence structure resembles that of the fractional Brownian motion (for H<1/2), but their sample paths are highly irregular (nowhere bounded with probability 1). Generalizations using different shapes of pulses are also discussed.

Recent work has shown that extreme doubly stochastic measures are supported on sets that have no axial cycles. We give a new proof of this result and examine the supporting set structure more closely. It is shown that the property of no axial cycles leads to a tree-like structure which naturally partitions the support into a collection of disjoint graphs of functions from the x-axis to the y-axis and from the y-axis to the x-axis. These functions are called a limb numbering system. It is shown that if the disjoint graphs in the limb numbering system are measurable, then the supporting set supports a unique doubly stochastic measure. Further, the limb structure can be used to develop a general method for constructing sets which support a unique doubly stochastic measure.

Let (X_i:n, Y_[i:n]), 1≤i≤n, denote the n pairs obtained by ordering a random sample of size n from an absolutely continuous bivariate population on the basis of X sample values. Here Y_[i:n] is called the concomitant of the ith order statistic. For 1≤k≤n, let V₁=max{{Y_[n-k+1:n],...,Y_[n:n]} and V₂=max{Y_[1:n],...,Y_[n-k:n]}. In this paper, we discuss the finite-sample and asymptotic joint distribution of (V₁,V₂). The asymptotic results are obtained when k=[np], 0<p<1, and when k is held fixed, as n→∞. We apply our results to the bivariate normal population and indicate how they can be used to determine k such that V₁ is close to Y_n:n, the maximum of the values of Y in the sample.

Most often the likelihood function based on discrete observations of a diffusion process is unknown, and estimators alternative to the well-behaved maximum likelihood estimator must be found. Traditionally, such estimators are defined with origin in the theory for continuous observation of the diffusion process, and are as a consequence strongly biased unless the discrete observation time-points are close. In contrast to these estimators, an estimator based on an approximation to the (unknown) likelihood function was proposed in Pedersen (1994). We prove consistency and asymptotic normality of this estimator with no assumptions on the distance between the discrete observation time-points.

Distributional assumptions for the returns on the underlying assets play a key role in valuation theories for derivative securities. Based on a data set consisting of daily prices of the 30 DAX shares over a three-year period, we investigate the distributional form of compound returns. After performing a number of statistical tests, it becomes clear that some of the standard assumptions cannot be justified. Instead, we introduce the class of hyperbolic distributions which can be fitted to the empirical returns with high accuracy. Two models based on hyperbolic Lévy motion are discussed. By studying the Esscher transform of the process with hyperbolic returns, we derive a valuation formula for derivative securities. The result suggests a correction of standard Black-Scholes pricing, especially for options close to expiration.