The Compound binomial model revisited

In this paper we re-cap the discrete model and views by Gerber (1988), also re-taken by other authors. That is, we consider a discrete time risk model where the aggregate claim process is compound binomial. In each period there is a claim with probability p, or no claim with probability 1 .p and that there is an independence in claim occurrrence in different time periods. The sequence of individual claims is a sequence of i.i.d. random variables, distributed on the positive integers, and independent of the claim number process. We follow the model formulation presented by Gerber (1988) and Dickson (1994).

Following the approach by Egídio dos Reis (2002) for the classical model, starting from a non-negative integer initial surplus, there is a positive probability that the risk process is ruined, i.e., it drops to negative values. If ruin occurs, it happens at the instant of a claim, then we can address the ruin probability problem, either finite or infinite time, by considering the number of claims necessary to get ruined. We then consider the calculation of the distribution of the number of claims up to ruin, if it occurs. Besides, since the process once ruined will recover to positive levels some time in the future with probability one, we also consider the distribution of the number of claims occurring during the recovery time period.

An interesting result is achieved concerning the particular case when the initial surplus is zero, which is the fact that the two discrete random variables above have the same distribution and that the distribution belongs to the Lagrangian-type family, and a closed form for the distribution is found.

From that, it is possible to find a recursion that allows the computation of the distribution of the number of claims up to ruin, considering any positive integer initial surplus. For these cases, we also find a formula for the distribution of the number of claim during a recovery time period.

Besides, with this model we will be able to compute approximations for the related quantities in the classical compound Poisson risk model.

Keywords: Ruin theory; compound binomial model; claim number up to ruin; claim number up to recovery; time to ruin; recursive calculation.