The risk models with non-local Poisson processes

The Poisson process is the most commonly used point process in modelling counting phenomena [21]. Even if the counting process has non-stationary increments, it can be shown to converge to the Poisson process if observed sufficiently long after a transient period as long as it constitutes a renewal process [43]. As such, it is important to review the key characteristics of the Poisson process as it serves as the main building block of more complex models. In the first part of this thesis, we propose two fractional risk models, where the classical risk process is time-changed by the mixture of tempered stable inverse subordinators. We characterise the risk processes by deriving the marginal distributions and establish the corresponding moments and covariance structure. In the second part of this thesis, we study the main characteristics of these models such as ruin probability and time of ruin, and illustrate the results with Monte Carlo simulations. The data suggests that the time of ruin can be approximated by the inverse gaussian distribution and its generalisations