Alexander Yu. Veretennikov

On large deviations in the averaging principle for SDE’s with a ``full dependence’’, revisited

We establish a large deviation principle for stochastic ndifferential equations with averaging in the case when all ncoefficients of the fast component depend on the slow one, nincluding diffusion.

Exponential Convergence of Degenerate Hybrid Stochastic Systems with Full Dependence

This research stems from a control problem for a suspension device. For a general class of switching stochastic mechanical systems (including closed-loop control ones), we establish the following: (1) existence and uniqueness of a weak solution and its strong Markov property, (2) mixing property in the form of the local Markov–Dobrushin condition, and (3) exponentially fast convergence to the unique stationary distribution. These results are proved for discontinuous coefficients under nondegenerate disturbances in the force field; for (3) a stability condition is additionally imposed. Linear growth of coefficients is allowed.

Ergodic Markov Processes and Poisson Equations (Lecture Notes)

These are lecture notes on the subject defined in the title. As such, they do not pretend to be really new, probably except for the only section about Poisson equations with potentials. Yet, the hope of the author is that they may serve as a bridge to the important area of Poisson equations in the whole space and with a parameter, the latter theme not being presented here. Why this area is so important was explained in many papers and books (see the references [12, 34, 35]): it provides one of the main tools in diffusion approximation in the area stochastic averaging. Hence, the aim of these lectures is to prepare the reader to real Poisson equations -- i.e., for differential operators instead of difference operators -- and, indeed, to diffusion approximation. Among other presented topics we mention coupling method.

On Polynomial Bounds of Convergence for the Availability Factor

A computable estimate of the readiness coefficient for a standard binary-state system is established in the case where both working and repair time distributions possess heavy tails.

An HJB Approach to a General Continuous-Time Mean-Variance Stochastic Control Problem

A general continuous mean-variance problem is considered for a diffusion controlled process where the reward functional has an integral and a terminal-time component. The problem is transformed into a superposition of a static and a dynamic optimization problem. The value function of the latter can be considered as the solution to a degenerate HJB equation either in viscosity or in Sobolev sense (after a regularization) under suitable assumptions and with implications with regards to the optimality of strategies. There is a useful interplay between the two approaches -- viscosity and Sobolev.