Professor Alexander Veretennikov
- Position: Professor
- Areas of expertise: stochastic differential equations & approximations; markov processes, stochastic control; mixing rates, averaging, diffusion approximation; large deviations; parameter estimations; mean-field models.
- Email: A.Veretennikov@leeds.ac.uk
- Phone: +44(0)113 343 5183
- Website: Personal webpage
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 with the ultimate goal to describe a class of processes, apparently with certain ergodic properties, for which filtering remains robust to certain errors in the model.
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 recent collaborator for this topic was 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 for a long time was S. Anulova, with whom we now started a new project related to ergodic control.
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. It is planned to collaborate with a stochastic group in University of Edinburgh studying similar problems, as well as with the INRIA group at Sophia-Antipolis, France, leaded by D. Talay.
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, exponential and slower than exponential have been and will be studied. One of my collaborators in this area is G. Zverkina. Most recent advances involve mean-field queueing models, which makes this direction close to McKean-Vlasov equations.
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