On the simulation of stochastic processes and noise driven synchronisation models

Abstract

We investigate the impact of noise with heavy tails on sychronisation within coupled systems of oscillators, the most classic case being called the `Kuramoto model'. The initial part of this work focuses on methods of simulation of heavy-tailed stochastic processes that are then implemented in the various synchronisation models discussed throughout the thesis. In particular, we explore the accept-rejection sampling method for stable and tempered stable Levy processes and the Wiener-Hopf Monte-Carlo algorithm for the simulation of the so-called beta-family of Levy processes. The first application of these heavy-tailed stochastic processes in an extension of an adversarial 'Blue vs Red' Kuramoto model from the deterministic and Gaussian cases to the cases where the models are driven by stable and tempered stable Levy noise. We apply techniques from Bayesian optimisation and the so-called 0-1 test for chaos to numerically investigate the impact of the stochastic noise under various parameterisations of the driving stochastic noise. This 'Blue vs Red' model is then extended into a 'combat-coordination model' through the development of an adversarial attrition model with spatial dependency driven by tempered stable Levy noise. We then consider extending our previously considered simulation techniques to the case of simulating conditioned stochastic processes, specifically processes conditioned to stay positive and conditioned on the terminal value. These simulation techniques for conditioned stochastic processes are then applied in the context of a conditioned Kuramoto model with a focus on the impact of the underlying network structure. We then advance an approach to mathematically represent networked human decision makers by adapting a Kuramoto model under heave-tailed noise and applying data from a recent study of military headquarters staff. We finally investigate pairs trading and portfolio optimisation under a general stochastic synchronisation model.