s-stable laws on Hilbert spaces, associated with some nonlinear transformations, were introduced by Jurek.((16, 18)) Here, we interpret certain s-stable motions as limits of total amount of claims processes (up to a deterministic reserve) of a portfolio of (nontraded) excess-of-loss reinsurance contracts and show that they lead to Erlang's model. We also give explicit formulas for the price of perpetual American options in case the logarithm of the price of the underlying asset is an s-stable motion. Furthermore, we generalize the concept of s-stability to simply connected nilpotent Lie groups. For step 2-nilpotent Lie groups we characterize the Levy measure and the s-domain of attraction of nongaussian s-stable convolution semigroups.