Ruin probabilities and aggregrate claims distributions for shot noise Cox processes

Abstract

We consider a risk process R _t where the claim arrival process is a superposition of a homogeneous Poisson process and a Cox process with a Poisson shot noise intensity process, capturing the effect of sudden increases of the claim intensity due to external events. The distribution of the aggregate claim size is investigated under these assumptions. For both light-tailed and heavy-tailed claim size distributions, asymptotic estimates for infinite-time and finite-time ruin probabilities are derived. Moreover, we discuss an extension of the model to an adaptive premium rule that is dynamically adjusted according to past claims experience.

Keywords:

• adative premium rule
• adjustment coefficient
• convex ordering
• Cramér-Lundberg approximation
• exponential change of measure
• Gärtner-Ellis theorem
• large deviations
• phase-type distribution
• saddlepoint approximation
• subexponential distribution

This research was carried out while the first author was visiting the Department of Mathematical Sciences of the University of Aarhus. He would like to thank the department for its hospitality. Supported by the Austrian Science Fund Project P-18392.