Self-similar processes in collective risk theory

A self-similar, continuous process with stationary increments is considered as an approximation to the surplus process in collective risk theory. This approximation can be seen as the weak limit of risk processes with linear premium income, where the claim sizes show a long-range dependence. It is then proved that the corresponding ruin times converge weakly to the ruin time of the approximation process. A situation where long-range dependence of claim sizes occurs is given in an example where the risk process evolves according to an environmental process with two states. If at least one of the distributions of the time between two changes of the state has a regularly varying tail, a long-range dependence is observed. By way of example, upper and lower bounds for the ruin probability of fractional Brownian motion with drift are obtained. Numerical calculations show, however, that these bounds are far from the true value.