First hitting time of uncertain random renewal reward process and its application in insurance risk process

Abstract

The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. This paper studies a type of renewal reward processes with random inter-arrival times and uncertain rewards from the point of view of first hitting time. The analytic expressions of the chance distribution and the expected value of the first hitting time are derived, and a numerical method for calculating the chance distribution is designed based on the Monte-Carlo simulation. Besides, the concept of first hitting time is applied to the insurance risk process and is employed to model the ruin index of an insurance company.