Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

Abstract

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival.We focus on the probability that this bivariate reserve process survives indefinitely. The infinite-horizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with ‘reflecting’ boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process.Under assumptions on the arrival process and the claim amounts, and using Wiener–Hopf factorization with one parameter, we explicitly determine the Laplace–Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution.Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.