# School of Mathematics

## Prof Alexander Yu Veretennikov

Professor of Statistics

Statistics

### Contact details

Room: 8.29

Tel: +44 (0)113 3435183

Email: A.Veretennikov @ leeds.ac.uk

### Keywords

Stochastic differential equations and approximation methods

Markov processes, stochastic control

Mixing rates, averaging, diffusion approximation

Large deviations

Parameter estimations in Markov models

Poisson equations, Green's functions estimates

Filtering for unspecified Markov models

Ergodicity in queueing theory

## Research interests

A long-standing problem in filtering with wrong initial data was solved in general state spaces in a series of my recent joint works with M.Kleptsyna. The problem was to find out whether an error in the initial distribution may be forgotten in the long run for a general filtering model. This direction remains one of my main interests for the near future.

Mixing was one of my main areas for a long time. A new version of local Markov-Dobrushin's condition has been proposed recently for so called Vaserstein's coupling. My current collaborator for this topic is O.Butkovsky. The topic also includes mixing for approximations of SDEs started about ten years ago in several joint papers with S.Klokov. Suppose a stochastic differential equation has a solution with some mixing rate, and, instead of the exact solution, one wishes to use its Euler's approximations. Is it correct that those approximations would have a similar mixing rate? In several important cases, conditions sufficient for establishing similar mixing bounds have been established. The topic also includes degenerate diffusions and jump-diffusion and hybrid processes or, in other terminology, diffusion with switching. My main collaborator in this direction is S.Anulova.

Recently, in a series of joint papers with E.Pardoux, the problem of Poisson equations "in the whole space" was studied. The problem is a highly important tool in the area of averaging and diffusion approximation where my current collaborator is A.Kulik.

McKean-Vlasov stochastic equations is a joint project with Yu.Mishura. It includes further studies of existence, uniqueness and asymptotic properties for this important class of processes investigated much less than "usual" Ito's equations.

Originally, Erlang's problem was stated in 1909 as a problem of a "customer loss" in a stationary regime for telephone systems; if a stationary regime has been computed, then the problem is solved. In practice, such solution may have little sense without rate of convergence to stationary regime. New classes of systems which admit convergence and mixing slower than exponential have been and will be studied.

### Useful links

Personal home page

http://maths.dept.shef.ac.uk/magic/course.php?id=205

http://www1.maths.leeds.ac.uk/~veretenn/projects2011.htm

## Current postgraduate students

Zeyu He (2013)

## Publications

**Veretennikov AY, Veretennikova EV** On partial derivatives of multivariate Bernstein polynomials *Siberian Advances in Mathematics*, **26**, 294-305, 2016

DOI:10.3103/S1055134416040039

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**Veretennikov A** On recurrence and availability factor for single–server system with general arrivals *Reliability: Theory&Applications*, **11**, 49-58, 2016

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**Bogachev VI, Shaposhnikov SV, Veretennikov AY** Differentiability of solutions of stationary fokker-planck-kolmogorov equations with respect to a parameter *Discrete and Continuous Dynamical Systems- Series A*, **36**, 3519-3543, 2016

DOI:10.3934/dcds.2016.36.3519

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**Kleptsyna ML, Veretennikov AY** On robustness of discrete time optimal filters *Mathematical Methods of Statistics*, **25**, 207-218, 2016

DOI:10.3103/S1066530716030042

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**Da Prato G, Flandoli F, Röckner M, Veretennikov AY** Strong uniqueness for sdes in hilbert spaces with nonregular drift *Annals of Probability*, **44**, 1985-2023, 2016

DOI:10.1214/15-AOP1016

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**Abu-Shanab R, Veretennikov AY** On asymptotic Borovkov-Sakhanenko inequality with unbounded parameter set *Theory of Probability and Mathematical Statistics*, **90**, 1-12, 2015

DOI:10.1090/tpms/945

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**Anulova SV, Veretennikov AY** Exponential convergence of multi-dimensional stochastic mechanical systems with switching impacts *Doklady Mathematics*, **91**, 60-63, 2015

DOI:10.1134/S1064562415010184

**Bogachev VI, Veretennikov AY, Shaposhnikov SV** Differentiability of invariant measures of diffusions with respect to a parameter *Doklady Mathematics*, **91**, 76-79, 2015

DOI:10.1134/S106456241501024X

**Veretennikov AY** On the rate of convergence for infinite server Erlang-Sevastyanov's problem *Queueing Systems*, **76**, 181-203, 2014

DOI:10.1007/s11134-013-9384-4

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**Veretennikov AY, Kulik AM** Diffusion approximation of systems with weakly ergodic markov perturbations. I *Theory of Probability and Mathematical Statistics*, **87**, 13-29, 2014

DOI:10.1090/S0094-9000-2014-00901-1

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**Veretennikov AY, Kulik AM** Diffusion approximation of systems with weakly ergodic markov perturbations. II *Theory of Probability and Mathematical Statistics*, **88**, 1-17, 2014

DOI:10.1090/S0094-9000-2014-00915-1

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**Veretennikov AY, Zverkina GA** Simple Proof of Dynkin's Formula for Single-Server Systems and Polynomial Convergence Rates *MARKOV PROCESSES AND RELATED FIELDS*, **20**, 479-504, 2014

**Veretennikov AY** On the rate of convergence for infinite server Erlang-Sevastyanov's problem *Queueing Systems*, 1-23, 2014

DOI:10.1007/s11134-013-9384-4

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**Veretennikov A** On large deviations in the averaging principle for sde's with a "full dependence," revisited *Discrete and Continuous Dynamical Systems - Series B*, **18**, 523-549, 2013

DOI:10.3934/dcdsb.2013.18.523

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**Klokov SA, Veretennikov AY** ON LOCAL MIXING CONDITIONS FOR SDE APPROXIMATIONS *THEORY OF PROBABILITY AND ITS APPLICATIONS*, **57**, 110-131, 2013

DOI:10.1137/S0040585X97985777

**Veretennikov A** On large deviations in the averaging principle for SDE’s with a ``full dependence’’, revisited *Discrete and Continuous Dynamical Systems ser. B*, **18**, 523-549, 2013

DOI:10.3934/dcdsb.2013.18.523

**Veretennikov A, Butkovsky O** On asymptotics for Vaserstein coupling of Markov chains *Stochastic Processes and their Applications*, **123**, 3518-3541, 2013

**Veretennikov A** On the rate of convergence to the stationary distribution in the single-server queuing system *Autom. Remote Control*, **74**, 1620-1629, 2013

**Veretennikov AY, Kulik AM** Extended Poisson equation for weakly ergodic Markov processes *Theory of Probability and Mathematical Statistics*, **85**, 23-39, 2012

DOI:10.1090/S0094-9000-2013-00871-0

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**Veretennikov A, Kleptsyna M** On filtering with unspecified initial data for nonuniformly ergodic signals *The Oxford Handbook of nonlinear filtering*, 267-298, 2011

**Veretennikov A** On Sobolev solutions of Poisson equations in R^d with a parameter *Journal of Mathematical Sciences*, **179**, 48-79, 2011

**Veretennikov A; Butkovsky O ** *On asymptotics of Vaserstein's coupling for a Markov chain*Proceedings (ISBN-13: 978-5-901158-15-9) 2011

**Veretennikov A; Anulova S ** *On ergodicity of highly degenerate hybrid stochastic control systems*Proceedings (ISBN-13: 978-5-901158-15-9) 2011

**Veretennikov AY** On the rate of beta-mixing and convergence to a stationary distribution in continuous-time Erlang-type systems *PROBL INFORM TRANSM+*, **46**, 382-389, 2010

DOI:10.1134/S0032946010040083

**Kleptsyna ML, Veretennikov AY** On discrete time ergodic filters with wrong initial data, 2 *STOCHASTICS*, **82**, 25-40, 2010

DOI:10.1080/17442500902723559

**Aivaliotis G, Veretennikov AY** On Bellman's equations for mean and variance control of a Markov diffusion *STOCHASTICS*, **82**, 41-51, 2010

DOI:10.1080/17442500902723567

**Veretennikov A; Anulova S ** *On ergodic properties of degenerate hybrid stochastic control systems*CDC, 2292-2297 2010

DOI:10.1109/CDC.2010.5717412

**Veretennikov A, Abourashchi N** On exponential mixing and rate of convergence for Student processes *Theory of Probability and Mathematical Statistics*, **81**, 1-13, 2010

**Veretennikov A, Abourashchi N** On stochastic averaging and mixing *Theory of Stochastic Processes*, **16(32)**, 111-130, 2010

**Veretennikov AY** On mixing rate and convegence to stationary regime in discrete time Erlang problem *AUTOMAT REM CONTR+*, **70**, 1992-2002, 2009

DOI:10.1134/S0005117909120078

**Veretennikov A, Abourashchi N** On exponential mixing bounds and convergence rate for reciprocal Gamma diffusion processes *Mathematical Communications*, **14**, 331-339, 2009

**Veretennikov AY, Kleptsyna ML** ON CONTINUOUS TIME ERGODIC FILTERS WITH WRONG INITIAL DATA *THEOR PROBAB APPL+*, **53**, 269-300, 2009

DOI:10.1137/S0040585X97983535

**Veretennikov A, Kleptsyna M** On discrete time filters with wrong initial data *Probability Theory and Related Fields*, **141**, 411-444, 2008

**Kleptsyna ML; Veretennikov A ** On discrete time ergodic filters with wrong initial data . *Authored under the alias of Veretennikov, A.Yu.*, 2007

DOI:10.1007/s00440-007-0089-7

**Kleptsyna ML, Veretennikov AY** On ergodic filters with wrong initial data *CR MATH*, **344**, 727-731, 2007

DOI:10.1016/j.crma.2007.04.015

**Klokov SA, Veretennikov AY** On mixing and convergence rates for a family of Markov processes approximating SDEs *Random Operators and Stochastic Equations*, **14**, 103-126, 2006

DOI:10.1163/156939706776953151

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**Veretennikov AY ** *On ergodic measures for McKean-Vlasov stochastic equations*Monte Carlo and Quasi-Monte Carlo Methods 2004, 471-486 2006

**Veretennikov AY ** *On lower bounds for mixing coefficients of Markov diffusions*From Stochastic Calculus to Mathematical Finance: The Shiryaev Festschrift, 623-633 2006

**Pardoux E, Veretennikov A** On the Poisson equation and diffusion approximation 3 *Annals of Probability*, **33**, 1111-1133, 2005

DOI:10.1214/009117905000000062

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**Veretennikov AY, Klokov SA** Mixing rates for the Euler scheme for Stochastic difference equations *DOKL MATH*, **69**, 273-274, 2004

**Klokov SA, Veretennikov AY** On subexponential mixing rate for Markov processes *THEOR PROBAB APPL+*, **49**, 110-122, 2004

DOI:10.1137/S0040585X97980841

**Pardoux E, Veretennikov A** On Poisson equation and diffusion approximation 2 *Annals of Probability*, **31**, 1166-1192, 2003

DOI:10.1214/aop/1055425774

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**Veretennikov AY** On large deviations for approximations of SDEs *PROBAB THEORY REL*, **125**, 135-152, 2003

DOI:10.1007/s00440-002-0235-1

**Veretennikov AY** On SDE and semigroup approximations and large deviations *THEOR PROBAB APPL+*, **47**, 733-741, 2002

DOI:10.1137/S0040585X97980075

**Veretennikov A** Coupling method for Markov chains under integral Doeblin condition *Theory of Stochastic Processes
Theory of Stochastic Processes*, **8**, 384-391, 2002

**Liptser R, Spokoiny V, Veretennikov A** Freidlin-Wentzell type moderate deviations for smooth processes *Markov Processes and Related Fields*, **4**, 2002

**Veretennikov A** On large deviations for approximations of SDEs on the torous *Theory of Probability and Its Applications*, **47**, 772-780, 2002

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**Veretennikov A** Essay on the Boris Vladimirovich Gnedenko theorem *Stochastic Processes and Their Applications*, **8**, 384-391, 2002

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**Hodges SD, Roberts G, Papaspiliopoulos O, Sentana E, Bingham NH, Cox DR, Nicolato E, Venardos E, Critchley F, Davis MHA, Tompkins R, Benth FE, Karlsen KH, Reikvam K, Brockwell PJ, Davis RA, Christensen BJ, Dellaportas P, McCoy EJ, Stephens DA, Diebold FX, Fruhwirth-Schnatter S, Genon-Catalot V, Laredo C, Grange CWJ, Griffin JE, Steel MFJ, Hobson D, Jensen JL, Jones MC, Lawrance AJ, Ledford AW, Leonenko NN, Levendorskii S, Mandelbrot BB, Meddahi N, Pitt MK, Priestley MB, Renault E, Rosinski J, Sato K, Taylor SJ, Tong H, Yang H, Veretennikov AY, Walker SG, Werker BJM, Wood A** Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics - Discussion *J ROY STAT SOC B*, **63**, 208-241, 2001

**Veretennikov A** On Polynomial mixing estimates for stochastic differential equations with a gradient drift *Theory of Probability and Its Applications*, **45**, 160-163, 2001

DOI:10.1137/S0040585X97978099

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**Veretennikov A** On the Poisson equation and diffusion approximation *Annals of Probability*, **29**, 1061-1085, 2001

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**Veretennikov A** On parameter estimation for 'polynomial ergodic' Markov chains with polynomial growth loss functions *Markov Processes and Related Fields*, **8**, 127-144, 2001

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**Veretennikov A** On large deviations for SDEs with small diffusion and averaging *Stochastic Processes and Their Applications*, **89**, 69-79, 2000

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**Veretennikov A** On polynomial mixing and convergence rate for stochastic difference and differential equations *Theory of Probability and Its Applications*, **44**, 361-374, 1999

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**Veretennikov A** On large deviations in the averaging principle for SDE's with a "full dependence" *Annals of Probability*, **27**, 284-296, 1999

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**Priouret P, Veretennikov A** A remark on the stability of the l.m.s. tracking algorithm *Stochastic Analysis and Applications*, **16**, 118-128, 1998

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**Pardoux E, Veretennikov A** Averaging of backward stochastic differential equations, with application to semilinear PDE's. *Stochastics and Stochastics Reports*, **60**, 255-270, 1997

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**Veretennikov A** On polynomial mixing bounds for stochastic differential equations *Stochastic Processes and Their Applications*, **70**, 115-127, 1997

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**Veretennikov A** On Poisson equations with a potential in the whole space for "ergodic'' generators *Theory of Probability and Mathematical Statistics*

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**Veretennikov A** Ergodic Markov processes and Poisson equations (lecture notes) *Springer Proceedings in Mathematics and Statistics*

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