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 E(Y2)<∞$\mathbb{E}(Y^2)<\mathrm{\infty }$$\mathbb{E}(Y^2)<\mathrm{\infty }$, the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto(α,θ)$(\alpha ,\theta )$ with $\alpha \le 2$. 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 $0<\alpha \le 2$. 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 $J_1$-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.

经典的 Cramér-Lundberg 风险过程通常用于对保险公司的盈余进行建模；它通过复合泊松过程以及恒定的保费收入率来描述索赔到达过程和索赔规模随机变量Y。什么时候E (是2) <无穷大$\mathbb{乙}（{是}^2）<\mathrm{无穷大}$$\mathbb{E}(Y^2)<\mathrm{\infty }$，该过程可以通过扩散过程来近似，但该要求消除了许多重尾索赔模型，例如帕累托( α , θ )$(\alpha ,\theta )$和$\alpha \le 2$。在本文中，我们通过假设Y位于α稳定随机变量的吸引力域中来推广众所周知的扩散近似，对于$0<\alpha \le 2$。然后，我们构建了一系列经典的 Cramér–Lundberg 风险过程，并表明该序列收敛于Skorokhod 中的α稳定 Lévy 运动$J_1$-拓扑。我们通过证明我们的过程的相应拉普拉斯指数的逐点收敛来证明这种收敛，据我们所知，这是一个新结果。为了应用这个收敛结果，我们展示了指数巴黎破产的 Gerber-Shiu 分布序列的收敛性，并且我们展示了障碍股息策略的收益函数序列的收敛性。这两个应用程序都提供了所述限制的新证明。