On a two-dimensional risk model with time-dependent claim sizes and risky investments

Consider a two-dimensional risk model, in which two insurance companies divide between them the claims in some specified proportions. Suppose that the claim sizes and inter-arrival times form a sequence of independent and identically distributed random pairs, with each pair obeying a dependence structure, and the surpluses of the two companies are invested into portfolios whose returns follow two different geometric Lévy processes. When the claim-size distribution is extended-regularly-varying tailed, asymptotic expressions for the ruin probability of this two-dimensional risk model are exhibited. Some numerical results are also presented to illustrate the accuracy of our asymptotic formulae.