Modeling of an insurance system and its large deviations analysis

Abstract

We model an insurance system consisting of one insurance company and one reinsurance company as a stochastic process in R 2 . The claim sizes { X i } are an iid sequence with light tails. The interarrival times { τ i } between claims are also iid and exponentially distributed. There is a fixed premium rate c 1 that the customers pay; c < c 1 of this rate goes to the reinsurance company. If a claim size is greater than R the reinsurance company pays for the claim. We study the bankruptcy of this system before it is able to handle N number of claims. It is assumed that each company has initial reserves that grow linearly in N and that the reinsurance company has a larger reserve than the insurance company. If c and c 1 are chosen appropriately, the probability of bankruptcy decays exponentially in N . We use large deviations (LD) analysis to compute the exponential decay rate and approximate the bankruptcy probability. We find that the LD analysis of the system decouples: the LD decay rate γ of the system is the minimum of the LD decay rates of the companies when they are considered independently and separately. An analytical and numerical study of γ as a function of ( c , R ) is carried out.