Abstract
In this paper, we consider a two-dimensional risk process in which the companies split each claim and premium in a fixed proportion. It serves as a classical framework of a quota-share reinsurance contract for a given business line. Such a contract reduces the insurer’s exposure to the liabilities created through its underwriting activities. For the analyzed model, we derive a joint infinite-time ruin probability formula for exponentially distributed claims. To this end, we apply a change of measure technique. We illustrate the admissible range of parameters of the risk process. We also justify our result using Monte Carlo simulations and compare it with Theorem 2 in Avram, Palmowski and Pistorius [Insurance: Mathematics and Economics 42 (2008) 227], which was obtained by explicitly inverting a Laplace transform of the ruin probability. Our formula leads to a correction of that result. Finally, we note that the obtained formula leads to efficient approximation of the ruin probability for other claim amount distributions using De Vylder’s idea.
Highlights
IntroductionPublisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations
Academic Editor: Anatoliy SwishchukReceived: 19 February 2021Accepted: 25 April 2021Published: 3 May 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Insurance companies have to comply with Solvency II requirements
We investigated the problem of ruin probability for a special case of the two-dimensional Cramér–Lundberg risk process
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Insurance companies have to comply with Solvency II requirements This framework offers insurers the possibility to improve business strategy and capital allocation. The model was introduced in Avram et al (2008b) and Avram et al (2008a) It induces a specific strong dependence between the two risk processes R1 (t) and R2 (t). A similar two-dimensional model, where both claims and pure premiums are split between the two companies, was considered by Ji and Robert (2018) but with a fractional Brownian motion as driving aggregate loss amount process, whereas Michna (2020) investigated a model driven by a general spectrally positive or negative Lévy process, see Avram et al (2008b).
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