Adaptive efficient analysis for big data ergodic diffusion models

We consider drift estimation problems for high dimension ergodic diffusion processes in nonparametric setting based on observations at discrete fixed time moments in the case when diffusion coefficients are unknown. To this end on the basis of sequential analysis methods we develop model selection procedures, for which we show non asymptotic sharp oracle inequalities. Through the obtained inequalities we show that the constructed model selection procedures are asymptotically efficient in adaptive setting, i.e. in the case when the model regularity is unknown. For the first time for such problem, we found in the explicit form the celebrated Pinsker constant which provides the sharp lower bound for the minimax squared accuracy normalized with the optimal convergence rate. Then we show that the asymptotic quadratic risk for the model selection procedure asymptotically coincides with the obtained lower bound, i.e this means that the constructed procedure is efficient. Finally, on the basis of the constructed model selection procedures in the framework of the big data models we provide the efficient estimation without using the parameter dimension or any sparse conditions

Renewal theory and its applications : lectures notes for the course "Stochastic modelling" taken by most Mathematics students and Economics students (directions of training 01.03.01 - Mathematics and 38.04.01 - Economics)

The goal of the course is to study the main tools of the renewal theory and their applications to some problems of the actuarial analysis for insurance companies in the framework of the Cremer - Lundberg models. We consider such important problems in the renewal theory as limit theorems for the renewal process and the ruin problems for the insurance companies with investments in the stochastic financial markets. The notes areintended for students of the Mathematics and Economics Faculties

Quickest change-point detection in time series with unknown distributions

We consider a problem of sequential detection o

Approximate hedging problem with transaction costs in stochastic volatility markets

This paper studies the problem of option replication in general stochastic volatility markets with transaction costs, using a new specification for the volatility adjustment in Leland's algorithm. We prove several limit theorems for the normalized replication error of Leland's strategy, as well as that of the strategy suggested by Lépinette. The asymptotic results obtained not only generalize the existing results, but also enable us to fix the underhedging property pointed out by Kabanov and Safarian. We also discuss possible methods to improve the convergence rate and to reduce the option price inclusive of transaction costs

Model selection method for efficient signals processing from discrete data

The paper considers the problem of robust adaptiv

Adaptive robust efficient methods for periodic signal processing observed with colours noises

In this paper, we consider the problem of robust adaptive efficient estimating a periodic signal observed in the transmission channel with the dependent noise defined by non-Gaussian Ornstein-Uhlenbeck processes with unknown correlation properties. Adaptive model selection procedures, based on the shrinkage weighted least squares estimates, are proposed. The comparison between shrinkage and least squares methods is studied and the advantages of the shrinkage methods are analyzed. Estimation properties for proposed statistical algorithms are studied on the basis of the robust mean square accuracy defined as the maximum mean square estimation error over all possible values of unknown noise parameters. Sharp oracle inequalities for the robust risks have been obtained. The robust efficiency of the model selection procedure has been established

Sequential analysis and its applications

This course is devoted to the main problems of the sequential analysis: sequential estimation and sequential hypothesis testing. Firstly we construct the least squares estimate for the scalar regression model and then we propose the sequential least squares estimate for the autoregression models. Finally, we study the non-asymptotic properties for the sequential estimation procedures. Then in the second part of this course we construct and study the sequential Wald procedure for hypothesis testing. We study its main properties: the mean times and the optimality properties in the sense of minimal mean time. Then we consider some examples of the Wald procedures. The notes are intended for students of the Mathematical Faculties