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Approximating the classical risk process by stable Lévy motion.
- Source :
-
Scandinavian Actuarial Journal . Sep2023, Vol. 2023 Issue 7, p679-707. 29p. - Publication Year :
- 2023
-
Abstract
- 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 (Y 2) < ∞ , the process can be approximated by a diffusion process, but that requirement eliminates many heavy-tailed claim models, such as the Pareto (α , θ) with α ≤ 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 < α ≤ 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *LEVY processes
*POISSON processes
*RANDOM variables
*EXPONENTS
*DIVIDENDS
Subjects
Details
- Language :
- English
- ISSN :
- 03461238
- Volume :
- 2023
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Scandinavian Actuarial Journal
- Publication Type :
- Academic Journal
- Accession number :
- 164582090
- Full Text :
- https://doi.org/10.1080/03461238.2022.2142157