Analysis of the multiple roots of the Lundberg fundamental equation in the PH (n) risk model

作者：Ji Lanpeng^*; Zhang Chunsheng

来源：Applied Stochastic Models in Business and Industry, 2012, 28(1): 73-90.

DOI：10.1002/asmb.899

• PH (n) risk model
• multiple roots
• eigenvectors
• linear independence
• Lundberg fundamental equation
• Gerber-Shiu penalty function

In the study of the Sparre Andersen risk model with phase-type (n) inter-claim times (PH (n) risk model), the distinct roots of the Lundberg fundamental equation in the right half of the complex plane and the linear independence of the eigenvectors related to the Lundberg matrix Ld(s) play important roles. In this paper, we study the case where the Lundberg fundamental equation has multiple roots or the corresponding eigenvectors are linearly dependent in the PH (n) risk model. We show that the multiple roots of the Lundberg fundamental equation det[Ld(s)] = 0 can be approximated by the distinct roots of the generalized Lundberg equation introduced in this paper and that the linearly dependent eigenvectors can be approximated by the corresponding linearly independent ones as well. Using this result we derive the expressions for the GerberShiu penalty function. Two special cases of the generalized Erlang(n) risk model and a Coxian(3) risk model are discussed in detail, which illustrate the applicability of main results. Finally, we consider the PH(2) risk model and conclude that the roots of the Lundberg fundamental equation in the right half of the complex plane are distinct and that the corresponding eigenvectors are linearly independent.

• 出版日期2012-2
• 单位南开大学