| Summary: | This thesis considers one of the central topics in the actuarial mathematics literature, deriving the probability of ruin in the collective risk model. The classical risk model and renewal risk models are focused in this project, where the claim number processes are assumed to be Poisson counting processes and any general renewal counting processes, respectively. The first part of this project is about the classical risk model. We look at the case when claim sizes follow a gamma distribution. Explicit expressions for ruin probabilities are derived via Laplace transform and inverse Laplace transform approach. The second half is about the renewal risk model. Very general assumptions on inter-arrival times are possible for the renewal risk model, which includes the classical risk model, Erlang risk model and fractional Poisson risk model. A new family of differential operators are de ned in order to construct the fractional integro-differential equations for ruin probabilities in such renewal risk models. Through the characteristic equation approach, specific fractional differential equations for the ruin probabilities can be solved explicitly, allowing for the analysis of the ruin probabilities.
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