A conditioned continuous-state branching process with applications

Abstract

A supercritical CB-process conditioned on explosion is again a Markov process. We characterize the transition semigroup of the conditioned process by its Laplace transform and by a Doob’s h-transform. The conditioned CB-process is constructed as the strong solution of a stochastic integral equation. We use the distribution of the conditioned process to construct directly the canonical Kuznetsov measure of the CB-process. The later is reconstructed from positive paths picked up by a Poisson random measure based on the canonical Kuznetsov measure.