On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions

Abstract

In this paper, we further extend the insurance risk model in Albrecher et al. (2011b), who proposed to only intervene in the compound Poisson risk process at the discrete time points ${L_k}_{k=0}^infty$ where the event of ruin is checked and dividend decisions are made. In practice, an insurance company typically balances its books (and monitors its solvency) more frequently than deciding on dividend payments. This motivates us to propose a generalization in which ruin is monitored at ${L_k}_{k=0}^infty$ whereas dividend decisions are only made at ${L_{jk}}_{k=0}^infty$ for some positive integer $j$. Assuming that the intervals between the time points ${L_k}_{k=0}^infty$ are Erlang($n$) distributed, the Erlangization technique (e.g. Asmussen et al. (2002)) allows us to model the more realistic situation with the books balanced e.g. monthly and dividend decisions made e.g. quarterly or semi-annually. Under a dividend barrier strategy with the above randomized interventions, we derive the expected discounted dividends paid until ruin. Numerical examples about dividend maximization with respect to the barrier $b$ and/or the value of $j$ are given.