Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims

Yang Yang; Kam C. Yuen; Jun-Feng Liu

doi:10.3934/jimo.2017044

Article Contents

2018, Volume 14, Issue 1: 231-247. Doi: 10.3934/jimo.2017044

This issue Previous Article Neutral and indifference pricing with stochastic correlation and volatility Next Article A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints

Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims

1.
Institute of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China
2.
Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, China
3.
Institute of Statistics and Data Science, Nanjing Audit University, Nanjing 211815, China

* Corresponding author: Yang Yang
* Corresponding author: Yang Yang

Received: December 31, 2015

Revised: January 31, 2017

Early access: March 2017

Published: January 2018

The research of Yang Yang was supported by the National Natural Science Foundation of China (No. 71471090, 71671166, 11301278), the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 14YJCZH182), Natural Science Foundation of Jiangsu Province of China (No. BK20161578), the Major Research Plan of Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 15KJA110001), Qing Lan Project, PAPD, Program of Excellent Science and Technology Innovation Team of the Jiangsu Higher Education Institutions of China, 333 Talent Training Project of Jiangsu Province, High Level Talent Project of Six Talents Peak of Jiangsu Province (No. JY-039), Project of Construction for Superior Subjects of Mathematics/Statistics of Jiangsu Higher Education Institutions, and Project of the Key Lab of Financial Engineering of Jiangsu Province (No. NSK2015-17). The research of Kam C. Yuen was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU17329216), and the CAE 2013 research grant from the Society of Actuaries. The research of Jun-feng Liu was supported by the National Natural Science Foundation of China (No. 11401313), and Natural Science Foundation of Jiangsu Province of China (No. BK20161579).

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Abstract

Consider a bivariate Lévy-driven risk model in which the loss process of an insurance company and the investment return process are two independent Lévy processes. Under the assumptions that the loss process has a Lévy measure of consistent variation and the return process fulfills a certain condition, we investigate the asymptotic behavior of the finite-time ruin probability. Further, we derive two asymptotic formulas for the finite-time and infinite-time ruin probabilities in a single Lévy-driven risk model, in which the loss process is still a Lévy process, whereas the investment return process reduces to a deterministic linear function. In such a special model, we relax the loss process with jumps whose common distribution is long tailed and of dominated variation.

Keywords:

Lévy-driven risk model,
finite-time and infinite-time ruin probabilities,
consistent variation,
dominated variation,
long tail,
asymptotics.

Mathematics Subject Classification: Primary: 91B30, 60G51; Secondary: 60K05.

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