On the probability of ruin of a joint-stock insurance company in the sparre Andersen risk model (Q2662925)

The paper develops the well-known Sparre Andersen risk model, taking into account the relevant fact that the insurance company can be a joint-stock company; moreover calculating the probability of ruin cannot disregard the consideration of dividend payments.In particular, the surplus of the insurance company paying dividends can be defined as $U_t = X_t-D_t$, where $Dt$ is the amount paid to shareholders as dividends at time t. Dividends are paid by the company in accordance with a specified barrier strategy.Within this framework, a barrier level $b_t$ is introduced: no dividends are paid when $Ut < bt$; once the surplus reaches the barrier $b_t$, the incoming premium starts to be paid (completely or partially) as dividends, and then the surplus remains on the barrier until the next claim occurs; finally, when $U_t > b_t$, then the total amount $U_t-b_t$ is paid as dividends.In this perspective, assuming that the intervals between the claim times have a gamma distribution, the paper provides an upper bound for the ruin probability.