Vadim Shcherbakov

Lecturer in Statistics
Department of Mathematics
Royal Holloway, University of London
Egham, Surrey, TW20 0EX, The United Kingdom
Tel: +44 1784 276528
vadim.shcherbakov@rhul.ac.uk

Research interests: probability theory and stochastic processes

Current research topics:

PhD projects: I am interested in supervising a PhD project related to my current research topics
and would be happy to receive applications from PhD candidates.

Example of PhD project: "Long term behaviour of interacting birth-and-death processes."
A single birth-and-death process on the set of non-negative integers is a classical probabilistic model for the size of a population. This is a continuous time Markov chain (CTMC) which evolves as follows. When the process is at state k, it can jump either to state k+1 (interpreted as birth event), or to state k-1, if k>0 (interpreted as death event), with transition rates that are state-dependent. The project is devoted to the long term behaviour of a class of Markov processes that can be interpreted as a system of birth-and-death processes, whose components evolve subject to a certain interaction (interacting birth-and-death processes). Originally interacting birth-and-death processes were motivated by modelling competition between populations. In this case they are known as competition processes, which is a class of population probabilistic models. Another interesting case of interecating birth-and-death processes is a growth process motivated by physical phenomenon known as cooperative sequential adsorption (CSA). In CSA duffusing particles can get adsorbed by a material surface, when they hit it. The main peculiarity of CSA is that the adsorbed particles can change the adsorption properties of the material in a sense that they either attract, or repulse other particles. The growth process is a system of pure birth processes whose components evolve subject to an interaction which is similar to that of CSA. In other words, the components of a growth process can either accelerate, or slow down the growth of each other. In the discrete time setting a growth process can be regarded as an interacting urn model. The latter is a class of random processes with reinforcement closely related to the generalised Polya urn model (another classical probabilistic model). The long term behaviour of a single birth-and-death process is well known. Given a set of transition rates one can, in principle, determine whether the corresponding Markov chain is (positive) recurrent or (explosive) transient and compute various other characteristics of the process. The long term behaviour of interacting birth-and-death processes is much less known. Studying this behaviour requires a solid background in probability and stochastic processes. A PhD student will learn and apply various probabilistic techniques. For example, this includes the Lyapunov function method (based on the martingale theory) and using similarity of the dynamics of interacting birth-and-death processes with that of branching processes.

Recent publications and preprints (for my other publications see mathscinet.ams.org or pure.royalholloway.ac.uk )

  1. V.Shcherbakov and S.Volkov (2020).
    Linear competition processes and generalised Polya urns with removals.
    arXiv:2001.01480 [math.PR]

  2. M.Menshikov and V.Shcherbakov (2019).
    Localisation in a growth model with interaction. Arbitrary graphs.
    arXiv:1903.04418[math.PR]

  3. S.Janson, V.Shcherbakov and S.Volkov (2019).
    Long term behaviour of a reversible system of interacting random walks.
    Journal of Statistical Physics, v.175, N1, pp.71-96.

  4. V.Shcherbakov and S.Volkov (2019).
    Boundary effects in competition processes.
    Journal of Applied Probability, v.56, N3, pp. 750-768.

  5. M.Costa, M.Menshikov, V.Shcherbakov and M.Vachkovskaia (2018).
    Localisation in a growth model with interaction.
    Journal of Statistical Physics, v.171, N6, pp.1150-1175.

  6. M.Menshikov and V.Shcherbakov (2018).
    Long term behaviour of two interacting birth-and-death processes.
    Markov Processes and Related Fields, v.24, N1, pp. 85-102. arXiv:1605.09607 [math.PR]

  7. A.Gairat and V.Shcherbakov (2017).
    Density of Skew Brownian motion and its functionals with applications in finance.
    Mathematical Finance, v.27, Issue 4, pp.1069-1088.

  8. V.Shcherbakov and A.Yambartsev (2016).
    A note on scaling limits for truncated birth-and-death processes with interaction.
    arXiv:1606.09061 [math.PR]

  9. V.Shcherbakov and S.Volkov (2015).
    Long term behaviour of locally interacting birth-and-death processes.
    Journal of Statistical Physics, v.158, N1, pp.132-157.