Volume 18, Issue 1

Abstract. The aim of the paper is to establish a convergence theorem for multi-dimensional stochastic approximation when the “innovations” satisfy some “light” averaging properties in the presence of a pathwise Lyapunov function. These averaging assumptions allow us to unify apparently remote frameworks where the innovations are simulated (possibly deterministic like in quasi-Monte Carlo simulation) or exogenous (like market data) with ergodic properties. We propose several fields of applications and illustrate our results on five examples mainly motivated by finance.

Abstract. We performed an extensive simulation study to compare the relative performance of many price-jump indicators with respect to false positive and false negative probabilities. We simulated twenty different time series specifications with different intraday noise volatility patterns and price-jump specifications. The double McNemar non-parametric test ( Psychometrika 12 (1947), 153–157) has been applied on constructed artificial time series to compare fourteen different price-jump indicators that are widely used in the literature. The results suggest large differences in terms of performance among the indicators, but we were able to identify the best-performing indicators. In the case of false positive probability, the best-performing price-jump indicator is based on thresholding with respect to centiles. In the case of false negative probability, the best indicator is based on bipower variation.

Abstract. Polynomial lattice point sets (PLPSs) (of rank 1) are special constructions of finite point sets which may have outstanding equidistribution properties. Such point sets are usually required as nodes in quasi-Monte Carlo rules. Any PLPS is a special instance of a -net in base as introduced by Niederreiter. In this paper we generalize PLPSs of rank 1 to what we call then PLPSs of rank and analyze their equidistribution properties in terms of the quality parameter and the (weighted) star discrepancy. We show the existence of PLPSs of “good” quality with respect to these quality measures. In case of the (weighted) star discrepancy such PLPSs can be constructed component-by-component wise. All results are for PLPSs in prime power base . Therefore, we also generalize results for PLPSs of rank 1 that were only known for prime bases so far.