Time stochastic s-convexity of claim processes

The purpose of this paper is to study the conditions on a stochastic process under which the s-convex ordering and the s-increasing convex stochastic ordering between two random instants is transformed into a stochastic ordering of the same type between the states occupied by this process at these moments. In this respect, the present work develops a previous study by Shaked and Wong (1995) [Probability in the Engineering and Informational Sciences 9, 563–580]. As an illustration, we show that the binomial and the Poisson processes, commonly used in actuarial sciences to model the occurrence of insured claims, possess this remarkable property.