On the infinite-horizon probability of (non)ruin for integer-valued claims

Abstract

We consider a compound Poisson process whose jumps are modelled as a sequence of positive, integer-valued, dependent random variables, W1,W2,…, viewed as insurance claim amounts. The number of points up to time t of the stationary Poisson process which models the claim arrivals is assumed to be independent of W1,W2,…. The premium income to the insurance company is represented by a nondecreasing, nonnegative, real-valued function h(t) on [0,∞) such that limt→∞h(t) = ∞. The function h(t) is interpreted as an upper boundary. The probability that the trajectory of such a compound Poisson process will not cross the upper boundary in infinite time is known as the infinite-horizon nonruin probability. Our main result in this paper is an explicit expression for the probability of infinite-horizon nonruin, assuming that certain conditions on the premium-income function, h(t), and the joint distribution of the claim amount random variables, W1,W2,…, hold. We have also considered the classical ruin probability model, in which W1,W2,… are assumed to be independent, identically distributed random variables and we let h(t)=u + ct. For this model we give a formula for the nonruin probability which is a special case of our main result. This formula is shown to coincide with the infinite-horizon nonruin probability formulae of Picard and Lefèvre (2001), Gerber (1988), (1989), and Shiu (1987), (1989).