Discounted probabilities and ruin theory in the compound binomial model

The aggregate claims are modeled as a compound binomial process, and the individual claim amounts are integer-valued. We study f(x, y; u), the “discounted” probability of ruin for an initial surplus u, such that the surplus just before ruin is x and the deficit at ruin is y. This function can be used to calculate the expected present value of a penalty that is due at ruin, and, if it is interpreted as a probability generating function, to obtain certain information about the time of ruin. An explicit formula for f(x, y; 0) is derived. Then it is shown how f(x, y; u) can be expressed in terms of f(x, y; 0) and an auxiliary function h(u) that is the solution of a certain recursive equation and is independent of x and y. As an application, we use the asymptotic expansion of h(u) to obtain an asymptotic formula for f(x, y; u). In this model, certain results can be obtained more easily than in the compound Poisson model and provide additional insight. For the case u=0, expressions for the expected present value of a payment of 1 at ruin and the expected time of ruin (given that ruin occurs) are obtained. A discrete version of Dickson’s formula is provided.