Strong Solutions and Simulation Techniques of Hybrid Stochastic Differential Equations with Past-Dependent Switching

Just as ordinary and partial differential equations are used extensively in modelling, stochastic differential equations (SDEs) are used to model systems that have inherent randomness or rely on behaviours that are too complex to fully observe. SDEs have found widespread application in the fields of mathematical biology, ecology, dynamical systems modelling, and mathematical finance, amongst others. A criticism of SDEs is their rigidity - the stochastic nature of their solutions is limited to the random fluctuations of the driving signal, inhibiting their ability to naturally describe systems exposed to both discrete and continuous sources of noise. To remedy this issue, we work with a family of coupled stochastic integral equations called hybrid stochastic differential equations (HSDEs) . HSDEs are solved by a twocomponent process. The first evolves according to an SDE that depends on the C'urrent state of the second component, which is a pure jump process on a discrete state space. To expand the scope for modelling and achieve greater generality, we allow the pure jump process to have a countably-infinite state space, and its transitions to depend upon the past behaviour of the first component. In this thesis, we obtain strong solutions for a robust class of HSDEs driven by Levy processes, and investigate sample path properties of these solutions, such as the Markov property and moment bounds. As a product of this analysis, we derive a simulable algorithm that converges to these solutions in a strong sense, and calculate its convergence rate. The algorithm relies on assumptions on the type and rate of convergence of existing simulation algorithms of Levy-driven SDEs. We also require the pure jump process component to be uniformizable. We discuss when these assumptions are satisfied, and present an alternative, weaker algorithm under milder conditions. Finally, we introduce HSDEs driven by families of general semimartingales as a rich source of future work, and obtain the existence and uniqueness of strong solutions for a broad class of these processes. We close with a brief discussion detailing the circumstances under which the HSDE algorithms can be applied to the semimartingale-driven case.Thesis (M.Phil.) -- University of Adelaide, School of Mathematical Scienecs, 202