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## Cox process functional learning

This article addresses the problem of functional supervised classification of Cox process trajectories, whose random intensity is driven by some exogenous random covariable. The classification task is achieved through a regularized convex empirical risk minimization procedure, and a non asymptotic oracle inequality is derived. We show that the algorithm provides a Bayes-risk consistent classifier. Furthermore, it is proved that the classifier converges at a rate which adapts to the unknown regularity of the intensity process. Our results are obtained by taking advantage of martingale and stochastic calculus arguments, which are natural in this context and fully exploit the functional nature of the problem.

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.

Varying coefficient models are useful generalizations of parametric linear models. They allow for parameters that depend on a covariate or that develop in time. They have a wide range of applications in time series analysis and regression. In time series analysis they have turned out to be a powerful approach to infer on behavioral and structural changes over time. In this paper, we are concerned with high dimensional varying coefficient models including the time varying coefficient model. Most studies in high dimensional nonparametric models treat penalization of series estimators. On the other side, kernel smoothing is a well established, well understood and successful approach in nonparametric estimation, in particular in the time varying coefficient model. But not much has been done for kernel smoothing in high-dimensional models. In this paper we will close this gap and we develop a penalized kernel smoothing approach for sparse high-dimensional models. The proposed estimators make use of a novel penalization scheme working with kernel smoothing. We establish a general and systematic theoretical analysis in high dimensions. This complements recent alternative approaches that are based on basis approximations and that allow more direct arguments to carry over insights from high-dimensional linear models. Furthermore, we develop theory not only for regression with independent observations but also for local stationary time series in high-dimensional sparse varying coefficient models. The development of theory for local stationary processes in a high-dimensional setting creates technical challenges. We also address issues of numerical implementation and of data adaptive selection of tuning parameters for penalization.The finite sample performance of the proposed methods is studied by simulations and it is illustrated by an empirical analysis of NASDAQ composite index data.

The paper is focused on the problem of multi-class classification of composite (piecewise-regular) objects (e.g., speech signals, complex images, etc.). We propose a mathematical model of composite object representation as a sequence of independent segments. Each segment is represented as a random sample of independent identically distributed feature vectors. Based on this model and statistical approach we reduce the task to a problem of composite hypothesis testing of segment homogeneity. Several nearest-neighbor criteria are implemented, for some of them the well-known special cases (e.g., the Kullback-Leibler minimum information discrimination principle, the probabilistic neural network) are highlighted. It is experimentally shown that the proposed approach improves the accuracy when compared with contemporary classifiers.

In 1994 and 1998 F. Delbaen and W. Schachermayer published two breakthrough papers in which they proved continuous-time versions of the Fundamental Theorem of Asset Pricing. This is one of the most remarkable achievements in modern Mathematical Finance which led to intensive investigations in many applications of the arbitrage theory on a mathematically rigorous basis of stochastic calculus.

This book provides the reader with a detailed understanding of all necessary attributes in stochastic calculus that are required for applications of the theory of stochastic integration in Mathematical Finance, in particular, in the arbitrage theory. The exposition follows the traditions of the Strasbourg school.

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.