A Ray-Knight Theorem for Spectrally Positive Stable Processes

Abstract

We generalize a classical second Ray-Knight theorem to spectrally positive stable processes. It is shown that the local time processes are solutions of certain stochastic Volterra equations driven by Poisson random measure and they belong to a class of fully novel non-Markov branching processes, named as rough continuous-state branching processes. Also, we prove the weak uniqueness of solutions to the stochastic Volterra equations by providing explicit exponential representations of the characteristic functionals in terms of the unique solutions to some associated nonlinear Volterra equations.