Article
Low-rank diffusion matrix estimation for high-dimensional time-changed Levy processes
The estimation of the diffusion matrix Σ of a high-dimensional, possibly time-changed Levy process is studied, based on discrete observations of the process with a fixed distance. A low-rank condition is imposed on Σ. Applying a spectral approach, we construct a weighted least-squares estimator with nuclear-norm-penalisation. We prove oracle inequalities and derive convergence rates for the diffusion matrix estimator. The convergence rates show a surprising dependency on the rank of Σ and are optimal in the minimax sense for fixed dimensions. Theoretical results are illustrated by a simulation study.
We consider N-component synchronization models defined in terms of stochastic particle systems with special interaction. For general (nonsymmetric) Markov models we discuss phenomenon of the long time stochastic synchronization. We study behavior of the system in different limit situations related to appropriate changes of variables and scalings. For N = 2 limit distributions are found explicitly.
We discuss the efficiency of the quadratic bridge volatility estimator in comparison with Parkinson, Garman-Klass and Roger-Satchell estimators. It is shown in particular that point and interval estimations of volatility, resting on the bridge estimator, are considerably more efficient than analogous estimations, resting on the Parkinson, Garman-Klass and Roger-Satchell ones. © 2012 Elsevier B.V. All rights reserved.
In this paper, we introduce a principally new method for modelling the dependence structure between two L{\'e}vy processes. The proposed method is based on some special properties of the time-changed Levy processes and can be viewed as an reasonable alternative to the copula approach.
-
Comparative statistical properties of Parkinson, Garman-Klass, Roger-Satchell and bridge oscillation estimators are discussed. Point and interval estimations, related with mentioned estimators are considered. The advantages of statistical indicators of the Brownian bridge than the other estimators had been shown. The results allow to conclude that the growth of trend leads to a significant shift of the calculated value for the classical.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traffic is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the final node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a finite-dimensional system of differential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of differential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.