Limit theorems for the fractional nonhomogeneous Poisson process

Risk process with mixture of tempered stable inverse subordinators: Analysis and synthesis

We propose two fractional risk models, where the classical risk process is time-changed by the mixture of tempered stable inverse subordinators. We characterize the risk processes by deriving the marginal distributions and establish the moments and covariance structure. We study the main characteristics of these models such as ruin probability and time to ruin and illustrate the results with Monte Carlo simulations. The data suggest that the ruin time can be approximated by the inverse gaussian distribution and its generalizations.