Statistics and dynamics of conditioned random systems

This research aims at expanding our understanding of random dynamical systems, i.e., systems whose underlying evolution laws are driven by noise. A thorough understanding of such systems is crucial as many phenomena arising in finance, ecology or climate science cannot be characterised by deterministic dynamics (in the absence of noise). Indeed, the introduction of noise to classical systems can drastically change their dynamical behaviour. Although random systems are very popular due to their many applications, there is so far, a limited theoretical general understanding of their qualitative behaviour even for low-dimensional systems.

The main focus of this PhD will be on studying absorbed random systems, that is systems conditioned on not being killed by some rare event. This could for instance model, the evolution of a stock below a certain price (barrier option), the expansion of a crack before a material breaks down or the growth of a population before its extinction. Due to these varied applications, absorbed systems have recently attracted the interest of a very active community.

Aims and objectives:
The scope of this research is threefold:
1. Develop a theoretical framework for the study of the dynamic behaviour of absorbed systems by investigating the existence and relevance of tools such as conditioned Lyapunov exponents. Such objects serve to characterise some qualities of these systems such as predictability or chaos.
2. Study the long-term present and past statistics of systems conditioned on survival for some natural systems for which such results are not known.
3. Design and implement computational methods to simulate such systems to the end of demonstrating the practicality (and thus relevance) of the aforementioned tools.

Novelty of the research methodology:
This project builds on recent advances in the theory of conditioned random dynamical systems achieved with the supervisors and collaborators. The project will make use of state-of-the-art techniques developed in stochastic analysis, dynamical systems and ergodic theory. The research of this PhD is stirred by a growing interest in absorbed systems at top mathematical institutions.

The project has Dr Martin Rasmussen and Prof. Jeroen Lamb as supervisors and Matheus De Castro, Dennis Chemnitz and Dr Maximilian Engel as collaborators.

This research is also supported by the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Algorithms.

This project falls within the EPSRC Mathematical Sciences theme and the non-linear systems, numerical analysis, statistics and applied probability research areas.