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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
View abstract

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
View abstract

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
View abstract

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
View abstract

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
View abstract

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 chainProceedings (ISBN-13: 978-5-901158-15-9) 2011

Veretennikov A; Anulova S On ergodicity of highly degenerate hybrid stochastic control systemsProceedings (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 systemsCDC, 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
View abstract

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

Veretennikov AY On lower bounds for mixing coefficients of Markov diffusionsFrom 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
View abstract

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
View abstract

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