The static potential in QCD - a full Two-Loop Calculation

A full analytic calculation of the two-loop diagrams contributing to the static potential in QCD is presented in detail. Using a renormalization group improvement, the ``three-loop'' potential in momentum space is thus derived and the third coefficient of the $\beta$-function for the V-scheme is given. The Fourier transformation to position space is then performed, and the result is briefly discussed.Comment: LaTeX, 25 pages, 12 figures include

Estimating the quadratic covariation matrix from noisy observations: Local method of moments and efficiency

An efficient estimator is constructed for the quadratic covariation or integrated co-volatility matrix of a multivariate continuous martingale based on noisy and nonsynchronous observations under high-frequency asymptotics. Our approach relies on an asymptotically equivalent continuous-time observation model where a local generalised method of moments in the spectral domain turns out to be optimal. Asymptotic semi-parametric efficiency is established in the Cram\'{e}r-Rao sense. Main findings are that nonsynchronicity of observation times has no impact on the asymptotics and that major efficiency gains are possible under correlation. Simulations illustrate the finite-sample behaviour.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1224 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Algorithmic trading engines versus human traders – do they behave different in securities markets?

After exchanges and alternative trading venues have introduced electronic execution mechanisms worldwide, the focus of the securities trading industry shifted to the use of fully electronic trading engines by banks, brokers and their institutional customers. These Algorithmic Trading engines enable order submissions without human intervention based on quantitative models applying historical and real-time market data. Although there is a widespread discussion on the pros and cons of Algorithmic Trading and on its impact on market volatility and market quality, little is known on how algorithms actually place their orders in the market and whether and in which respect this differs form other order submissions. Based on a dataset that – for the first time – includes a specific flag to enable the identification of orders submitted by Algorithmic Trading engines, the paper investigates the extent of Algorithmic Trading activity and specifically their order placement strategies in comparison to human traders in the Xetra trading system. It is shown that Algorithmic Trading has become a relevant part of overall market activity and that Algorithmic Trading engines fundamentally differ from human traders in their order submission, modification and deletion behavior as they exploit real-time market data and latest market movements

Arithmetic properties of projective varieties of almost minimal degree

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X \subset {\mathbb P}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $\tilde {X} \subset {\mathbb P}^{r + 1}$ from an appropriate point $p \in {\mathbb P}^{r + 1} \setminus \tilde {X}$. We focus on the latter situation and study $X$ by means of the projection $\tilde {X} \to X$. If $X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring $B$ of the projecting variety $\tilde {X}$ is the endomorphism ring of the canonical module $K(A)$ of the homogeneous coordinate ring $A$ of $X.$ If $X$ is non-normal and is maximally Del Pezzo, that is arithmetically Cohen-Macaulay but not arithmetically normal $B$ is just the graded integral closure of $A.$ It turns out, that the geometry of the projection $\tilde {X} \to X$ is governed by the arithmetic depth of $X$ in any case. We study in particular the case in which the projecting variety $\tilde {X} \subset {\mathbb P}^{r + 1}$ is a cone (over a) rational normal scroll. In this case $X$ is contained in a variety of minimal degree $Y \subset {\mathbb P}^r$ such that \codim_Y(X) = 1. We use this to approximate the Betti numbers of $X$. In addition we present several examples to illustrate our results and we draw some of the links to Fujita's classification of polarized varieties of $\Delta$-genus 1.Comment: corrected, revised version. J. Algebraic Geom., to appea