Discontinuous Measure‐Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure‐valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure‐valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)‐valued vector measures.