We study the ruin probability where the claim sizes are modeled by a stationary ergodic symmetric α-stable process. We exploit the flow representation of such processes, and we consider the processes generated by conservative flows. We focus on two classes of conservative α-stable processes (one discrete-time and one continuous-time), and give results for the order of magnitude of the ruin probability as the initial capital goes to infinity. We also prove a solidarity property for null-recurrent Markov chains as an auxiliary result, which might be of independent interest.
