Approximating the classical risk process by stable Lévy motion

The classical Cramér–Lundberg risk process is commonly used to model the surplus of an insurer; it characterizes the claim arrival process and the claim size random variable Y through a compound Poisson process, along with a constant rate of premium income. When , the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto with . In this paper, we generalize the well known diffusion approximation by assuming that Y lies in the domain of attraction of an α-stable random variable, for . Then, we construct a sequence of classical Cramér–Lundberg risk processes and show that the sequence converges to an α-stable Lévy motion in the Skorokhod -topology. We prove this convergence by proving the pointwise convergence of the corresponding Laplace exponents of our processes, which to our knowledge, is a new result. To apply this convergence result, we show the convergence of a sequence of Gerber–Shiu distributions of exponential Parisian ruin, and we show the convergence of a sequence of payoff functions for barrier dividend strategies. Both of these applications provide new proofs of the stated limits.