### Book

## Modern problems of stochastic analysis and statistics - Selected contributions in honor of Valentin Konakov

This book brings together the latest findings in the area of stochastic analysis and statistics. The individual chapters cover a wide range of topics from limit theorems, Markov processes, nonparametric methods, acturial science, population dynamics, and many others. The volume is dedicated to Valentin Konakov, head of the International Laboratory of Stochastic Analysis and its Applications on the occasion of his 70th birthday. Contributions were prepared by the participants of the international conference of the international conference “Modern problems of stochastic analysis and statistics”, held at the Higher School of Economics in Moscow from May 29 - June 2, 2016. It offers a valuable reference resource for researchers and graduate students interested in modern stochastics.

The paper contains a complete analysis of the Galton–Watson models with immigration, including the processes in the random environment, stationary or nonstationary ones. We also study the branching random walk on $Z^d$ with immigration and prove the existence of the limits for the first two correlation functions.

In this paper we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way the complexity order of the standard Monte Carlo algorithm (ε−3) can be reduced down to ε−2 log(ε−1) in case of the Euler scheme with ε being the precision to be achieved. These theoretical results are illustrated by several numerical examples.

We consider a sequence of general filtered statistical models with a finite-dimensional parameter. It is tacitly assumed that a proper rescaling of the parameter space is already done (so we deal with a local parameter) and also time rescaling is done if necessary. Our first and main purpose is to give sufficient conditions for the existence of certain uniform in time linear–quadratic approximations of log-likelihood ratio processes. Second, we prove general theorems establishing LAN, LAMN and LAQ properties for these models based on these linear–quadratic approximations. Our third purpose is to prove three theorems related to the necessity of the conditions in our main result. These theorems assert that these conditions are necessarily satisfied if (1) an approximation of a much more general form exists and a (necessary) condition of asymptotic negligibility of jumps of likelihood ratio processes holds, or (2) we have LAN property at every moment of time and the limiting models are continuous in time, or (3) we have LAN property, Hellinger processes are asymptotically degenerate at the terminal times, and the condition of asymptotic negligibility of jumps holds.

In the first part of the paper we consider a "random flight" process in \(R^d\) and obtain the weak limits under different transformations of the Poissonian switching times. In the second part we construct diffusion approximations for this process and investigate their accuracy. To prove the weak convergence result we use the approach of Stroock and Varadhan (1979). We consider more general model which may be called "random walk over ellipsoids in \(R^d\)". For this model we establish the Edgeworth type expansion. The main tool in this part is the parametrix method (Konakov (2012), Konakov and Mammen (2009)).

Ergodic properties of Markov chains are studied, with the emphasis to convergence rate bounds and to applications to Poisson equations

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.

In this paper we prove that the spatially homogeneous Landau equation for Maxwellian molecules can be represented through the product of two elementary stochastic processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition. © 2015, University of Washington. All rights reserved.

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.