On joint ruin probability for a bidimensional Lévy-driven risk model with stochastic returns and heavy-tailed claims

In this paper we study the joint ruin problem for two insurance companies that divide between them the losses in positive proportions δ1δ1 and δ2δ2 (modeling an insurance and a re-insurance company). Assume that the surplus process of i  -th (i=1,2)(i=1,2) company UiUi has the form of dUi(t)=Ui(t−)dRi(t)−δidP(t)dUi(t)=Ui(t−)dRi(t)−δidP(t), t>0t>0, with Ui(0)=xi>0Ui(0)=xi>0, and P   and RiRi two Lévy processes representing, respectively, a loss process and a stochastic return process. Supposing that the loss process P   has a Lévy measure of consistent variation and the Laplace exponent of RiRi(i=1,2)(i=1,2) satisfies some conditions, an asymptotic estimate for this joint ruin problem is established.