Ruin probability approximation for a class of renewal processes with heavy tails

This paper examines an integro-differential equation of the survival probability d(u) for a class of risk processes in which claims occur as an ordinary renewal process. Specifically, according to the model proposed by Dickson, claims are assumed to occur as an Erlang process. We determine d(u) by using an exponential claim size distribution, thus comparing the results to those obtained from the classical Cramér-Lundberg model. After having transformed the integro-differential equation into an integral equation, we find an approximate solution for d(u). We then test it in exponential cases. Furthermore, we consider the case of large claims characterized by heavy-tailed claim size distributions. The approximate solution found for d(u) also applies to certain subexponential distributions. Finally, we obtain an expression for the asymptotic behaviour of when the claim size distribution has a regularly varying tail.