Stability and instability in stochastic models

This project is theoretical, however, with a clear opportunity of applications. A range of stochastic models covered by the problem include Markov chains, stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs), birth-death processes, telecommunication models, etc. Stability is a highly important feature of most of models used in practice. On the other hand, studying instability may help to find out limitation of stability and, hence, predict and tackle some undesirable consequences (disasters). Applications are possible to mathematical biology, queueing, service systems, mathematical finance, etc. A general objective is to find new classes of systems that show stability/instability features important in many practical applications. Possible options of the general problem may include Stability and diffusion approximations in random networks and queueing systems, Estimating mixing bounds for recurrent Markov processes, Controlled random networks and stochastic differential equations, Averaging in complex stochastic models, Parameter estimation for recurrent Markov chains/SDEs, Filtering for non-specified Markov models, etc.

Some prerequisites:

1. good knowledge of probability and stochastic processes (SDEs welcome);
2. analysis (ODEs and PDEs included), Lebesgue integration.

Some books to read (additional reading may be needed - not all books are required for a particular version of the programme)
1. A.N.Shiryaev, Probability.
2. H.M.Taylor, S.Karlin, A first/second course in stochastic processes.
4. W.Feller, Introduction to Probability, v.v. 1-2.
5. D.Stirzaker, Stochastic processes and models.
6. N.V.Krylov, Introduction to the theory of random processes.
7. Some (any) textbook on bifurcations.