Abstract
This paper considers the distribution of some extremum on the risk process whose income depend on the current reserve. We first construct the defective renewal sequence and obtain the density function of them. By the presented renewal measure and the strong Markov property, the distribution of the first hitting time is obtained explicitly. Then, the ruin probability and the probability that the surplus process less than x is obtained. Furthermore, the distribution of supreme profits before ruin, the joint distributions of the supreme profit and the deficit before the time of the surplus process first up-crossing level zero after ruin, and the joint distributions of the supreme profit and the deficit before the surplus process leave zero ultimately are derived. Finally, the exact calculating results for them are obtained when the individual claim amounts in the compound Poisson risk model are exponentially distributed.
Highlights
Before introducing the model, we revisit some important risk models
All these models and many other risk models modified from the compound Poisson risk model belong to the risk process whose income depend on the current reserve
The distribution of extremum is very important in risk theory, which can portray the best and worst condition of an insurance company and provide early warning for the development of the insurance company
Summary
We revisit some important risk models. Assume that the claim arrival process {N (t)}t≥0 is a Poisson process with parameter , and that the claim sizes {Zk }k≥1 independent of {N (t)}t≥0, are positive, independent and identically distributed random variables with common density function f and mean value μ. The risk models above have a common characteristic, that is, the surplus process moves according to the same differential equation in between jumps. In this paper we use Px(·) to denote P(·|x ∈ R) the probability of the process {U (t)}t≥0 with initial value x generated on ( , F∞). Das and Mahavier (2012) study the joint distribution of the surplus immediately before ruin and the deficit at ruin for the compound Poisson risk model with constant interest.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
