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

Abstract

a b s t r a c tWe 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 infinitehorizon 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.

A correction term for the covariance of renewal-reward processes with multivariate rewards

“…The renewal-reward process with multivariate rewards analysed here often plays a role as part of a more complicated stochastic model -for example in multidimensional risk models, such as in [2]. Our results may help analysis of such risk models, at least in some asymptotic regime.…”

Optimal dividend payments for a two-dimensional insurance risk process

“…For some recent papers considering dependent risks, see Dhaene and Goovaerts [19,20], Goovaerts and Dhaene [25], Müller [34,35], Denuit et al [17], Ambagaspitiya [3], Dhaene and Denuit [18], Hu and Wu [27] and Chan et al [14]. Ruin probability expressions for a two-dimensional risk process were also studied in Avram et al [6,7] for simultaneous claim arrivals and proportional claim sizes and recently in Badila et al [13] and Ivanovs and Boxma [29] in a more general framework.…”

Optimal dividend strategies for two collaborating insurance companies

“…Asmussen and Albrecher [3,Chapter XIII.9] for an overview. Ruin probability expressions for a twodimensional risk process were studied in [5], [6] for simultaneous claim arrivals and proportional claim sizes and recently in [11] and [17] in a more general framework. In [8], the problem of optimally transferring capital between two portfolios in the presence of transaction costs was considered; see also [10].…”