Final Activity Report Summary - LARDEV (Asymptotics of stochastic dynamical systems)
Stochastic dynamical systems are used to model various real-world phenomena. An important tool for studying stochastic dynamical systems is provided by asymptotic analysis, since the knowledge of limit laws for the characteristics of the system significantly enhances our ability to manage and utilise it more efficiently.
This project was concerned with developing asymptotic methods and obtaining results for stochastic systems that either involved discontinuous dependence of the dynamic on the system state or were governed by random variables with heavy-tailed distribution functions. Systems of this sort were ubiquitous in telecommunications, mathematical finance, network modelling etc. Specific mathematical models included random graphs, queuing systems and random walks. The research was mostly focused on large deviation asymptotics for queuing systems and random walks. A new approach for deriving large deviation asymptotics for a vast class of models with discontinuities was devised. It was quite general and enabled to tackle a variety of systems, such as generalised Jackson networks, 'join-the-shortest-queue' models, infinite server models with least-loaded routing, diffusion processes with discontinuous drift and others.
For random walks, a completely new type of asymptotics was discovered in the problem of reaching a high level by a random walk with negative drift. A related direction of research concerned investigating random graphs with preferential attachment. These models were of interest because they followed the power law for the node degrees which was observed in many real-world networks, such as the worldwide web and social interaction networks. A diffusion approximation limit theorem for the degree process of a graph with preferential attachment was obtained. Put together, the results significantly increased the arsenal of tools that were available for the study of stochastic systems.
This project was concerned with developing asymptotic methods and obtaining results for stochastic systems that either involved discontinuous dependence of the dynamic on the system state or were governed by random variables with heavy-tailed distribution functions. Systems of this sort were ubiquitous in telecommunications, mathematical finance, network modelling etc. Specific mathematical models included random graphs, queuing systems and random walks. The research was mostly focused on large deviation asymptotics for queuing systems and random walks. A new approach for deriving large deviation asymptotics for a vast class of models with discontinuities was devised. It was quite general and enabled to tackle a variety of systems, such as generalised Jackson networks, 'join-the-shortest-queue' models, infinite server models with least-loaded routing, diffusion processes with discontinuous drift and others.
For random walks, a completely new type of asymptotics was discovered in the problem of reaching a high level by a random walk with negative drift. A related direction of research concerned investigating random graphs with preferential attachment. These models were of interest because they followed the power law for the node degrees which was observed in many real-world networks, such as the worldwide web and social interaction networks. A diffusion approximation limit theorem for the degree process of a graph with preferential attachment was obtained. Put together, the results significantly increased the arsenal of tools that were available for the study of stochastic systems.