Non-Linear diffusion processes and applications

Diffusion models are useful tools for quantifying the dynamics of continuously evolving processes. Using diffusion models it is possible to formulate compact descriptions for the dynamics of real-world processes in terms of stochastic differential equations. Despite the exibility of these models, they can often be extremely difficult to work with. This is especially true for non-linear and/or time-inhomogeneous diffusion models where even basic statistical properties of the process can be elusive. As such, we explore various techniques for analysing non-linear diffusion models in contexts ranging from conducting inference under discrete observation and solving first passage time problems, to the analysis of jump diffusion processes and highly non-linear diffusion processes. We apply the methodology to a number of real-world ecological and financial problems of interest and demonstrate how non-linear diffusion models can be used to better understand such phenomena. In conjunction with the methodology, we develop a series of software packages that can be used to accurately and efficiently analyse various classes of non-linear diffusion models