Ruin probabilities in the presence of heavy-tails and interest rates

TLDR

The results apply to heavy‑tailed claim size distributions such as Pareto, log‑gamma, Benktander, and stable laws. The study investigates the infinite‑time ruin probability in a Cramér‑Lundberg model with interest on reserves. The authors analyze the model under a regularly varying heavy‑tailed claim size distribution, focusing on the large‑claims regime and incorporating a positive force of interest. With a positive interest rate, the ruin probability decays asymptotically as κδ (1−F(u)), differing from the non‑interest case where it decays as κ (1−F(u)).

Abstract

Abstract We study the infinite time ruin probability for the classical Cramér-Lundberg model, where the company also receives interest on its reserve. We consider the large claims case, where the claim size distribution F has a regularly varying tail. Hence our results apply for instance to Pareto, loggamma, certain Benktander and stable claim size distributions. We prove that for a positive force of interest δ the ruin probability ψδ (u) ∼ κδ (1 - F(u)) as the initial risk reserve u→∞. This is quantitatively different from the non-interest model, where ψ(u) ∼ κ (1 – F(y)) dy.