International conference «Modern problems of stochastic analysis and statistics»

In this talk we consider a semiparametric problem of estimating a sparse covariance matrix from the observations of the normal distribution with this matrix convoluted with some unknown error distribution.This is joint work with M. Trabs and A. Tsybakov.

It is well known that insurance is practiced at least three thousand years. However actuarial sciences investigating the insurance problems emerged more recently, namely, in the 17-th century due to Edmond Halley's mortality tables (1693) and Daniel Bernoulli's utility function notion (1738) in the next century. They started the so-called deterministic period of actuarial sciences development. The second period is probabilistic (or stochastic). It is characterized by introduction of stochastic models in life- and non-life insurance. The most important achievement of this period is the collective risk theory initiated by Filip Lundberg dissertation (1903) and later contributions by Harald Cram'er. The classical Cram\'er-Lundberg model and its various ramifications are still the base for the study of ruin probability of insurance company. The third (modern) period of actuarial sciences demonstrates the strong interaction with financial mathematics and utilization of sophisticated mathematical tools. The primary goal of insurance company is redistribution of risks and satisfaction of policyholders claims, hence, the popularity of reliability approach, that is, thorough analysis of ruin probability. Being a corporation it has a secondary but very important goal, namely, dividends payment to the shareholders. So, the alternative cost approach was started by De Finetti in 1957. Thus, there arose the new research directions in actuarial sciences specific for modern period. They include, along with dividends payments, reinsurance and investment problems. Hence, the treatment of complex models and consideration of new classes of processes, such as martingales, diffusion, L'evy processes or generalized renewal, is needed. Several types of objective functions and various methods are used to implement the stochastic models. It is also important to mention investigation of systems asymptotic behavior and their stability. However neither the list of problems nor references are complete, more details will be given during the talk.

In many stochastic models a response variable $Y$ depends on some factors collection $X=(X_1,\ldots,X_n)$. Such models arise, e.g., in medicine and biology where $Y$ characterizes the health state of a patient and $X$ contains genetic and nongenetic factors. The challenging problem is to identify the relevant set $(X_{i_1},\ldots,X_{i_r})$ of factors where $1\leq i_1\! < \ldots \!<i_r\leq n$ ($r\!<\!n$). We discuss various approaches to this problem and compare different tools. Statistical and machine learning methods are considered. In this regard we refer, e.g., to the recent book [J.Gareth, D.Witten, T.Hastie and R.Tibshirani. An Introduction to Statistical Learning with Applications in R. Springer, New York, 2014]. A number of exhaustive, stochastic and heuristic methods to detect epistasis (in genetics) are considered in [J.Shang, J.Zhang, Y.Sun, D.Liu, D.Ye and Y.Yin. Performance analysis of novel methods for detecting epistasis. Bioinformatics, 2012]. We concentrate on the multifactor dimensionality reduction (MDR) method introduced by M.Ritchie et al. and developed in a series of publications. New theorems concerning the asymptotic behavior of certain statistics involving cross-validation procedure are established. We tackle also the simulation problems related to variable selection.

We consider the moving particle process in $R^{d}$ which is defined in the following way. There are two independent sequences $\left( T_{k}\right) $ and $\left( \theta _{k}\right) $ of random variables. The first forms a Poisson point process in $R_{+}$ and the second is i.i.d sequence with common distribution concentrated on the unit sphere $S^{d-1}.$ The values $% \theta _{k}$ are interpreted as the directions, and $T_{k}$ as the moments of change of directions. A\ particle starts from zero and moves in the direction $\theta _{1}$ up to the moment $T_{1}.$ It then changes direction to $\theta _{2}$ and moves on within the time interval $T_{2}-T_{1},$ etc. The speed is constant at all steps. The position of the particle at time $t$ is denoted by $X(t).$ Study of the processes of this type has a long history. The first work dates back probably to Pearson (1905) and continued by Kluyer (1906) and Rayleigh (1919). B. Mandelbrot [B. Mandelbrot, The fractional geometry of Nature, N-Y (1982)] considered the case where the increments \\ $T_{n}-T_{n-1}$ form i.i.d. sequence with the common law having a heavy tail. He also introduced the term "Levy flights" later evolved into the "Random flights". To date, a large number of works were accumulated, devoted to the study of such processes, we mention here only articles of A. Kolesnik [A.D. Kolesnik, The explicit probability distribution of a six-dimensional random flight] and E. Orsingher [E. Orsingher, A. De Gregorio, Reflecting Random Flights, E.Orsingher, R.Garra, Random flighs governed by Klein-Gordon type partial differential equations, E.Orsingher, A. De Grigorio, Flying randomly in Rd with Dirichlet displacements] which contain an extensive bibliography and where for different assumptions on $\left( T_{k}\right) $ and $\left( \theta _{k}\right) $ the exact formulas for the distribution of $X(t)$ were derived. Our goals are different. Firstly, we are interested in the global behavior of the process $X=(X(t),t\in R_{+}),$ namely, we are looking for conditions under which suitably scaled and normalized process $X$ weakly converges in $C[0,1].$ Secondly, we want to construct diffusion approximations for the process $X$ and evaluate the accuracy of such approximations. It is clear that in the homogeneous case, the process $X$ is a conventional random walk, and then a limit process is the Brownian motion. In a non homogeneous case, it was possible to distinguish three modes that determine the type of limiting process. For a more precise description of the results it is convenient to assume that $T_{k}=f(\Gamma _{k}),$ where $% ( \Gamma _{k}) $ is a standard homogeneous Poisson point process on $R_{+}.$ If the function $f$ has power growth, $f(t)=t^{\alpha },\;\;\alpha \geq 1,$ the situation is analogous to the uniform case and then in the limit we obtain a Gaussian process which is a lineally transformed Brownian motion. In the case of exponential growth, $f(t)=e^{t\beta },$ $\beta >0,$ the limiting process is piecewise linear with an infinite number of units, but for all $ \epsilon >0$ the number of units in the interval $[\varepsilon ,1]$ will be a.s. finite. Finally, with the super exponential growth the limiting process degenerates: its trajectories are linear functions a.s. In the second part of the article the process $X$ is considered as a Markov chain. We construct diffusion approximations for this process and investigate their accuracy. The main tool is the parametrix method [V. Konakov. Metod parametriksa diya diffusii i cepei Markova].

In this talk, we present a parametrix expansion for a one dimensional elliptic killed diffusion. This allows to obtain the existence and to study the regularity properties of the density for the killed elliptic diffusion and its exit time. We also provide Gaussian upper estimates and derive a probabilistic representation that allows for the exact simulation of the related Markov semigroup. This is joint work with Arturo Kohatsu­Higa and Libo Li.

We study the asymptotic of the exit time of a Markov walk from the positive half-line under spectral gap assumptions. We also obtain the limit of the distribution of the walk conditioned that it stays positive. We shall illustrate our results by considering two examples: the products of random matrices in the general linear group and the affine random walks on the real line. The challenge is the proof of the existence of the harmonic function related to the underlying Markov transition operators defined on appropriately constructed Banach spaces. The asymptotics of the exit time and of the conditional law are transferred from the Gaussian model to the model under consideration with the help of a Komlos-Major-Tusnady type approximation result for Markov chains.

The aim of the talk is to describe the set of all possible joint distributions of terminal values of an integrable increasing process and its compensator. This result is adjacent to the results obtained by Rogers (1993) on the set of terminal values of a uniformly integrable martingale and its maximum. Our results clarify the structure of certain extreme distributions in the both problems.

Inhomogeneous random graph models encompass many network models such as stochastic block models and latent position models. In this work, we derive optimal rates of estimation of the probability matrix in these models. Our results cover the important setting of sparse networks. Nonparametric rates for graphon estimation are also derived when the probability matrix is sampled according to a graphon model. The results shed light on the differences between estimation under the empirical loss (the probability matrix estimation) and under the integrated loss (the graphon estimation). This is a joint work with A. Tsybakov and N. Verzelen.

The parametrix method has been used for many purposes and many different variations of it have been introduced in the past. In this talk, I will present some of the probabilistic interpretations and some of its applications that I have found interesting for me in the recent past.

We discuss recent trends in the Monge­-Kantorovich transportation theory, with an emphasis on applications, especially to functional and entropic inequalities. Our analysis will rely, in particular, on recent results on the structure of eigenvalues of the corresponding Monge­-Ampere equation.

In the framework of an abstract statistical model we discuss how to use the solution of one estimation problem (Problem A) in order to construct an estimator in another, completely different, Problem B. As a solution of Problem A we understand a data-driven selection from a given family of estimators $A(H)= \{\hat A_ h, h \in H \}$ and establishing for the selected estimator so-called oracle inequality. If $\hat h \in H $ is the selected parameter and $ B (H) = {\hat B_h, h \in H \}$ is an estimator's collection built in Problem B we suggest to use the estimator $\hat B_{\hat h}$. We present very general selection rule led to selector $\hat h$ and find conditions under which the estimator $\hat B_{\hat h}$ is reasonable. Our approach is illustrated by several examples related to adaptive estimation.

There are two main classes of classical (non-quantum) dynamics for N-particle systems. Namely, deterministic hamiltonian systems and interacting Markov processes. Formally speaking, the first one is a particular case of the second. However, deep results for both systems differ strongly in the extent and in the methods. This is mainly due to the fact that the classical theory of Markov processes developed independently of its deterministic counterpart, using the assumption that the conditional probability measures P some fixed measure on the state space X, for some fixed n and all x ∈ X, This condition greatly simplified the proofs of convergence to equilibrium in the ergodic case. It is even more important that any weakening of this condition on P gave rise to serious results and theories. For purely hamiltonian many particle dynamics one should first formalize the famous Ludwig Boltzmann hypothesis concerning convergence to one of the equilibrium measures - Liouville or Gibbs, or possibly to other measures. There are two evident possibilities for such formalization: 1) for given purely hamiltonian dynamics one should find a class of random initial conditions such that for any initial condition from this class there will be convergence to one of these measures, 2) find interaction of the given hamiltonian system with external world (reservoir) and prove convergence, for any initial conditions, to one (depending on the choice of the reservoir and on the interaction) of the equilibrium measures. We consider here the second possibility but our goal here is more challenging. Namely, we would like to find the minimal possible interaction with the external world which provides the desired convergence. It is big surprise that interaction of only one of N particles with extermal media is sufficient for this. We believe that it is what Boltzmann really meant. We give review and generalization of the (n) (x, .) are absolutely continuous with respect to results in this direction, and discuss some new and more general problems. Mathematical tools we need are quite numerous: linear and non- ̄linear analysis of hamiltonian many particle systems, techniques for Markov chains with continuum state space, random operator dynamics in Banach spaces. t

We study the weak error associated with the Euler scheme of some diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of H ̈older continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.

We consider a class of population models that includes not only migration and birth and death processes but also the immigration of particles to the system from outside. Under natural conditions, we prove the ergodicity of such models, i.e., 8 the existence of steady states. In the presence of immigration, the steady states are stable with respect to all random in space and time perturbations that are small enough, in contrast to the case of KPP-type models. For the steady states, we study also the effect of intermittency and prove several limit theorems.

We investigate micro­to­macroscopic derivations in two models of living cells, in presence to cell­cell adhesive interactions. We rigorously address two PDE­based models, one featuring non­local terms and another purely local, as a a result of a law of large numbers for stochastic particle systems, with moderate interactions in the sense of K. Oelshchlager. Talk is based on joint work with Dario Trevisan (arxiv.org/abs/1601.05241).

We consider the fractional equations of the form $ ( \frac{\partial ^2}{\partial t^2} -c^2 \frac{\partial^2}{\partial x^2})^a u(x,t)= \lambda u(x,t), 0 2.

We establish non­asymptotic Gaussian concentration bounds for the difference between the invariant measure of an ergodic Brownian diffusion process and the empirical distribution $\nu$ of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions $f$ such that $f­\nu(f)$ is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr\"obenius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable confidence intervals for the approximating scheme. As a theoretical application, we finally derive non­asymptotic deviation bounds for the almost sure Central Limit Theorem.

Let$\mathbf{X(}t)=(X_{1}(t),\ldots,X_{d}(t))$ be a Gaussian stationary vector process. Let $h:{\mathbb{R}}^{d}\rightarrow {\mathbb{R}}$ be a homogeneous function. We study probabilities of large extremes of the Gaussian chaos process $h(\mathbf{X}(t))$. Important examples of the processes are $h_{1}(\boldsymbol{X}(t))=\prod_{i=1}^{d}X_{i}(t)$ and $h_{2}(\boldsymbol{X}(t))=\sum_{i=1}^{d}a_{i}X_{i}^{2}(t)$. We give a review of the existing results, obtained partially in cooperation with E. Hashorva, D. Korshunov, and A. Zhdanov. We also describe the corresponding methods of studying, among them are the Laplace asymptotic method and asymptotic methods for probabilities of far excursions of Gaussian vector processes trajectories. As well, statements of some new problems in this direction are given.

The parametrix method is a general tool useful to construct fundamental solutions of linear partial differential equations with non-negative characteristic form. It also provides accurate upper bounds of the fundamental solutions. Analogous lower bounds can be proved by building ”Harnack chains”, that are sequences of points such that local Harnack inequalities hold. The optimization in the construction of Harnack chains gives accurate lower bounds of the fundamental solutions. In this note we discuss the construction of optimal Harnack chains for some specific examples of partial differential operators related to stochastic processes.

We consider the problem of approximation of a locally stationary random process with a variable smoothness index defined on an interval. An example of such function is a multifractional Brownian motion, which is an extension of the fractional Brownian motion with path regularity varying in time. Probabilistic models based on the locally stationary random processes with variable smoothness became recently an object of interest for applications in various areas (e.g., Internet traffic, financial records, natural landscapes) due to their flexibility for matching local regularity properties. Approximation of continuous and smooth random functions with unique singularity point is studied in [Abramowicz, K. and Seleznjev, O. (2011)]. Assuming that the smoothness index attains its unique minimum in the interval (an isolated singularity point), we propose a method for construction of observation points sets (sampling designs) in approximation (piecewise constant approximation, [Hashorva, E., Lifshits, M., and Seleznjev, O. (2015)]). For such methods, we find the exact asymptotic rate for the integrated mean square error. Further, we show that the suggested rate is optimal, e.g., convergence is faster than for conventional regular designs. The obtained results can be used in various problems in signal processing, e.g., in optimization of compressing digitized signals, in numerical analysis of random functions, e.g., in simulation studies with controlled accuracy for functionals on realizations of random processes.

Motivated by several applications in statistics, learning, and econometrics, we consider the following problem. Let \( M_{t} \) be a vector sub­Gaussian martingale in \( R^{p} \) with the quadratic characteristic \( _{t} \). The main result establishes a sharp deviation bound for norm \( \| _{t}^{­1/2} M_{t} \| \), the leading term is the same as in the Gaussian case, with an additional payment for variability of the quadratic characteristic \( _{t} \).

The main discussion will be on and around the well-known fact: Any probability distribution with finite moments is either M-determinate (uniquely determined by its moments), or it is M-indeterminate (non-unique). A brief and systematic account of classical and well-known results (Cramer, Carleman, Hausdorff, Krein) will be followed by presenting recent developments and new results (some are just published, others are in good progress). This includes: HardyÕs condition for M-determinacy; use of the rate of growth of the moments; Stieltjes classes; SDEs with prescribed moment properties; statistical inference problems. If time permits some challenging open questions will be outlined.

We present new sensitivity analyses w.r.t. the long-range/memory noise parameter for solutions to stochastic differential equations and the probability distributions of their first passage times at given thresholds. Here we consider the case of stochastic differential equations driven by fractional Brownian motions and the sensitivity, when the Hurst parameter H of the noise tends to the pure Brownian value, of probability distributions of certain functionals of the trajectories of the solutions $\{X^H_t\}_{t\in \mathbb{R}_+}$. We first present accurate sensitivity estimates w.r.t. H around the critical Brownian parameter $H=\tfrac{1}{2}$ of time marginal probability distributions of $X^H$. We second summarize our sensitivity analysis for the Laplace transform of first passage time of $X^H$ at a given threshold. Our technique required to get new accurate Gaussian estimates on the density of $X^H_t$ which we discuss at the end of the talk. The lecture is based on a joint work with Alexandre Richard (Inria).

We derive non-asymptotic bounds for the minimax risk of variable selection under the expected Hamming loss in the problem of recovery of s-sparse vectors in R^d whose non-zero components are greater than a > 0. We get exact expressions for the non-asymptotic minimax risk as a function of (d, s, a) and find explicitly the minimax selectors. Analogous results are obtained for the probability of wrong recovery of the sparsity pattern. As corollaries, we derive necessary and sufficient conditions for such asymptotic properties as almost full recovery and exact recovery. Moreover, we propose data-driven selectors that provide almost full and exact recovery adaptive to the parameters (s, a) of the classes. This is a joint work with Cristina Butucea and Natalia Stepanova.

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular itapplies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithmÕs performance with several examples. This is a joint work with Stefan Ankirchner and Thomas Kruse.