In this thesis, we study two classes of point processes by analysing key properties and discussing applications in finance and insurance. The first point process studied was the default indicator process in credit risk modelling. We considered a pure jump L ́evy process of finite variation for the asset value and an unobservable random barrier. The default time was defined as the first time the asset value falls below the barrier. Using the indistinguishable intensity process and the instantaneous likelihood process, we proved the absolute continuity of the compensator for the default indicator process, or equivalently, the existence of the intensity process of the default time. Moreover, we found the explicit representation of the intensity in terms of the distance between the asset value and its running minimal value, thus the intensity is an endogenous process, which sheds new light on the relationship between the intensity model and the structural model.
The second class of point processes is the Dynamic Contagion Process, which has intensities modelled with a shot-noise component describing the external impact and mutually-exciting jump components that describe the internal contagion effect. In the bivariate case, we found the stationarity condition with which we explored the diffusion approximation of the high frequency point process system and applied it in filtering. In the univariate case, we constructed a pure jump process derived from a dynamic contagion process and showed the weak convergence to a Cox-Ingersoll- Ross model (CIR) process. The pathwise approximation provides an alternative method of simulating the square-root processes and can be further extended to the approximation of the Heston model in option pricing.